Sequences in the OEIS which I have either commented on or added to

 

The OEIS is the best resource for integer sequences on the internet and is maintained and presided over by Neil Sloane with the assistance of an editorial panel of around 75 mathematicians and computer programmers. It currently contains around 200000 sequences, and those listed below are the ones that I have commented on:

 

 

1.

A013929

Numbers that are not squarefree. Comment added that this sequence is different from the sequence of numbers n such that A007913 (n)<phi(n).

 

 

 

2.

A016742

Even, square numbers. Sum to infinity of the reciprocals given as 1/24 x pi^2.

 

 

 

3.

A063990

Amicable numbers. Mathematica code added.

 

 

 

4.

A038547

Least number with exactly n odd divisors. Connection made with the least integers to have a politeness of 1, 2, 3, 4, … and requisite Mathematica code added.

 

 

 

5.

A145389

Digital roots of the triangular numbers. Considerably simplified closed form added.

 

 

 

6.

A006431

Numbers that have a unique partition into a sum of four squares of nonnegative integers. Definition amended from representations to partitions – as these are not the same. Further comments and amendments added later, including a third order recurrence relation which applies from the sixteenth term (96) onwards.

 

 

 

7.

A056992

Digital roots of the square numbers. Generating function added.

 

 

 

8.

A078644

tau(2n^2)^2. Comment added to the effect that a(n) is also the number of Pythagorean triangles with radius of the inscribed circle equal to n.

 

 

 

9.

A048242

Numbers that are not the sum of two not necessarily distinct abundant numbers. Four references to the literature added.

 

 

 

10.

A161005

 

Sum of amicable pairs. The author of the sequence acknowledges the use of my Mathematica code to calculate the terms.

 

 

 

11.

A093195

Least number which is the sum of two distinct squares of nonnegative integers in exactly n ways. Terms a(11) through to a(32) added together with an algorithm to compute them.

 

 

 

12.

A164775

a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.

Terms a(11) and a(12) added, together with an appropriate reference to the literature.

 

 

 

13.

A004441

Numbers that are not the sum of 4 distinct nonzero squares. Mathematica code added and a thirty fifth order recurrence relation identified.

 

 

 

14.

A004438

Numbers that are not the sum of 5 distinct squares. Mathematica code added.

 

 

 

15.

A001481

Numbers that are the sum of two squares of non-negative integers. Comment added regarding the structure of such numbers and a further comment added regarding their properties.

 

 

 

16.

A022544

Numbers that are not the sum of two squares. Mathematica code added, together with a comment to the effect that these are also the integers that have an equal number of 4k+1 and 4k+3 divisors.

 

 

 

17.

A000161

Number of partitions of n into sums of two squares (of non-negative integers). Mathematica code added and a formula given, together with a reference to Hirschhorn’s 2000 paper “Some formulae for partitions into squares”.

 

 

 

18.

A000446

Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways. Algorithm added to allow (relatively) easy computation of any member of this sequence.

 

 

 

19.

A008784

Integers that have a primitive partition into a sum of two squares. Comment added to the effect that if an integer is a member of this sequence then so too are all of its divisors, together with the Mathematica code required to generate the sequence. A reference to Dickson II was also added, which credits the statement and proof of this Theorem to Euler.

 

 

 

20.

A004215

Numbers that are the sum of four but no fewer non-zero squares. Improved Mathematica code added.

 

 

 

21.

A000164

Number of partitions of n into sums of three squares of non-negative integers. A somewhat remarkable formula added, together with the Mathematica code required to execute it. Some simpler mathematica code also added, together with a reference to Hirschhorn’s 2000 paper.

 

 

 

22.

A004433

Numbers that are the sum of four distinct nonzero squares. Mathematica code added.

 

 

 

23.

A002635

Number of partitions of n into sums of four squares of non-negative integers. Mathematica code added, together with a reference to Hirschhorn’s 2000 paper (which contains a somewhat complicated looking formula to generate the nth term of the sequence).

 

 

 

24.

A063954

“Of course every number is the sum of 4 squares; these are the odd numbers such that the first square can be taken to be any square < n”. Mathematica code added, together with a reference to Pall’s 1932 paper on sums of two and four squares.

 

 

 

25.

A004437

Numbers that are not the sum of 4 distinct squares. Comment added regarding the structure (and the properties) of the integers that are members of this sequence, together with a thirty first order recurrence relation and the Mathematica code required to generate it. Reference also included to Pall’s 1933 paper “On sums of squares”.

 

 

 

26.

A000415

Numbers that are the sum of two but no fewer nonzero squares. Comment added regarding the structure of the integers that are members of this sequence, together with the Mathematica code required to generate them. References also included to Guy’s 1994 paper “Every Number is Expressible as the Sum of How Many Polygonal Numbers?” and Grosswald’s 1985 book “Representation of Integers as Sums of Squares”.

 

 

 

27.

A145389

Digital roots of triangular numbers – generating function added for n>0.

 

 

 

28.

A008954

Units’ digits of triangular numbers – closed form added

 

 

 

29.

A042963

Congruent to 1 or 2 mod 4. Third order recurrence relation added, together with a closed form which applies if we regard the sequence as having offset 1. Also added the Mathematica code required to generate the sequence.

 

 

 

30.

A014493

Odd triangular numbers. Fifth order recurrence relation added, together with a closed form.

 

 

 

31.

A014601

Congruent to 0 or 3 mod 4. Improved Mathematica code added.

 

 

 

32.

A014494

Even triangular numbers. Fifth order recurrence relation added, together with a closed form and two different Mathematica routines required to generate the sequence.

 

 

 

33.

A002817

Doubly triangular numbers. Fifth order recurrence relation added and the existing Mathematica code edited (as the original was incomplete). Comment added to the effect that the members of this sequence are also A000217 (A000217 (n)).

 

 

 

34.

A138591

Sums of two or more consecutive integers. Comment regarding polite numbers corrected and improved Mathematica code added.

 

 

 

35.

A069283

-1+number of odd divisors of n. Comment added to the effect that a(n) is called the politeness of n. Two further comments added that give the structure of the members of this sequence with a reference to Tom Apostol’s 2003 paper “Sums of Consecutive Positive Integers”, a link to Wikepedia and some improved Mathematica code.

 

 

 

36.

A051533

Numbers which are the sum of two positive triangular numbers. Mathematica code added.

 

 

 

37.

A038547

Least number with exactly n odd divisors. Mathematica code added.

 

 

 

38.

A020756

Numbers which are the sum of two triangular numbers. Mathematica code added, together with three comments regarding the structure of the members of this sequence, a cross-reference to A000217 and a reference to Ewell’s 1992 paper “On Sums of Triangular Numbers and Sums of Squares”.

 

 

 

39.

A052343

Number of ways to write n as the unordered sum of two triangular numbers (zero allowed). A simple Mathematica code added, together with a formula to allow direct computation of any term in the sequence and the Mathematica code required to verify it.

 

 

 

40.

A008441

Number of (ordered) ways of writing n as the sum of 2 triangular numbers. Mathematica code added.

 

 

 

41.

A014637

Odd heptagonal numbers. Closed form added together with a fifth order homogeneous recurrence, a fourth order inhomogeneous recurrence and the Mathematica code required to generate the sequence.

 

 

 

42.

A008438

Sum of divisors of 2n+1. Efficient Mathematica code added.

 

 

 

43.

A007331

Fourier coefficients of E_{\infty,4}, where E_{\infty,4} is the unique normalized weight-4 modular form for \Gamma_0(2) with simple zeros at i*\infty. Mathematica code added.

 

 

 

44.

A020757

Numbers that are not the sum of two triangular numbers. Mathematica code added, together with a comment regarding the structure of the members of this sequence and a reference to Ewell’s 1992 paper “On Sums of Triangular Numbers and Sums of Squares”.

 

 

 

45.

A129445

Numbers k>0 such that k^2 is a centered triangular number. Reference added to Beldon and Gardiner’s 2002 paper “Triangular Numbers and Perfect Squares”.

 

 

 

46.

A005832

Numbers n such that n and 2n-1 are primes. Misdirected link removed.

 

 

 

47.

A120628

Primes such that their double is 1 away from a prime number. Mathematica code added.

 

 

 

48.

A029549

a(0) = 0, a(1) = 6, a(2) = 210; for n >= 0, a(n+3) = 35*a(n+2) - 35*a(n+1) + a(n). Two closed forms added (in terms of the ceiling and floor functions), together with a statement to the effect that the sum to infinity of the reciprocals of the terms of this sequence is 3-2sqrt(2). A comment also added that the terms of A165518 are generated by the same homogeneous recurrence relation as the terms of this sequence.

 

 

 

49.

A061455

Triangular numbers whose digit reversal is also a triangular number. Mathematica code added.

 

 

 

50.

A076713

Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits. Mathematica code added.

 

 

 

51.

A046194

Heptagonal triangular numbers. Added a fifth order homogeneous recurrence, two closed forms, a generating function and Mathematica code. Also, the long term behaviour of the ratio of consecutive terms was quantified precisely, according to the parity of n.

 

 

 

52.

A001751

The prime numbers and their doubles. Mathematica code added, together with a comment that relates the members of this sequence to those integers that can appear as a leg in precisely one

Pythagorean triple (both primitive and imprimitive).

 

 

 

53.

A059270

Numbers which are both the sum of n+1 consecutive integers and the sum of the n immediately higher consecutive integers. Improved Mathematica code added, together with a third order inhomogeneous recurrence relation, a fourth order homogeneous recurrence relation and a generating function.

 

 

 

54.

A039835

Indices of triangular numbers which are also heptagonal. Added a fifth order homogeneous recurrence, two closed forms and Mathematica code. Also, the long term behaviour of the ratio of consecutive terms was quantified precisely, according to the parity of n.

 

 

 

55.

A144396

The odd numbers greater than 1. Comment added to the effect that a(n) is the shortest leg of the nth Pythagorean triple with consecutive longer leg and hypotenuse, together with a formula for the n th such triple and cross references to the corresponding sequences of longer legs and hypotenuses.

 

 

 

56.

A046193

Indices of heptagonal numbers which are also triangular. Added a fourth order inhomogeneous recurrence, a fifth order homogeneous recurrence, two closed forms, a generating function and Mathematica code. Also, the long term behaviour of the ratio of consecutive terms was quantified precisely, according to the parity of n.

 

 

 

57.

A099799

a(n) = least integer that begins a run of exactly n consecutive integers that can be the hypotenuse of a Pythagorean triangle. Reference added to Shanks’ 1968 paper which reviews Beiler’s method of computing such integers (although not necessarily the least).

 

 

 

58.

A068068

Number of odd unitary divisors of n. d is a unitary divisor of n if d divides n and GCD(d,n/d)=1. Comment added to the effect that “a(n) is the total number of primitive Pythagorean triples satisfying area=n x perimeter, or equivalently 2 raised to the power of the number of distinct, odd primes contained in n”. Two references to the literature also added.

 

 

 

59.

A161462

Final digit of sum of all the natural numbers from n to 2*n-1. Alternative definition added along with a twentieth order homogeneous recurrence, a nineteenth order inhomogeneous recurrence, Mathematica code and a cross reference to the pentagonal numbers.

 

 

 

60.

A175734

The largest n-digit number with 3 divisors. Assistance given to a commentator acknowledged.

 

 

 

61.

A014632

Odd pentagonal numbers. Closed form and a fourth order inhomogeneous recurrence relation added.

 

 

 

62.

A014633

Even pentagonal numbers. Closed form added, together with a fourth order inhomogeneous recurrence and a fifth order homogeneous recurrence.

 

 

 

63.

A024352

Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c. Closed form added, together with a fourth order homogeneous recurrence.

 

 

 

64.

A001318

Generalised pentagonal numbers. Fourth order, inhomogeneous recurrence relation added. Also a comment added later (with justification) to the effect that this is a self-generating sequence that can be simply constructed from knowledge of the first term alone.

 

 

 

65.

A033934

(10^n+1)^2. Comment added to the effect that “The members of this sequence are both perfect squares and palindromes. We may deduce from this that there are infinitely many palindromic squares, or equivalently that A002779 is an infinite sequence”.

 

 

 

66.

A090466

Regular figurative or polygonal numbers of order greater than 2. Mathematica code corrected and link to content of A090428 noted.

 

 

 

67.

A002452

(9^n-1)/8. An existing comment made precise, improved Mathematica code added together with a second order homogeneous recurrence relation.

 

 

 

68.

A000384

Hexagonal numbers. Second order, inhomogeneous recurrence relation added. Also, an existing comment qualified to the effect that “if p1 and p2 are two arbitrarily chosen distinct primes then a(n) is the number of divisors of (p1^2*p2)^(n-1)" or equivalently of any term of A054753^(n-1).

 

 

 

69.

A046079

Number of Pythagorean triangles with leg n. Comment added to the effect that the position of the ones in this sequence (for n>2) corresponds to the prime numbers and their doubles.

 

 

 

70.

A046181

Indices of octagonal numbers which are also triangular. Added a fourth order inhomogeneous recurrence, a fifth order homogeneous recurrence, two closed forms, a generating function and Mathematica code. Also, the long term behaviour of the ratio of consecutive terms was given, according to the parity of n.

 

 

 

71.

A046182

Indices of triangular numbers which are also octagonal. Added a fourth order inhomogeneous recurrence, a fifth order homogeneous recurrence, two closed forms, a generating function and Mathematica code. Also, the long term behaviour of the ratio of consecutive terms was given, according to the parity of n.

 

 

 

72.

A046183

Octagonal triangular numbers. Added a fourth order inhomogeneous recurrence, a fifth order homogeneous recurrence, two closed forms and Mathematica code. Also, the long term behaviour of the ratio of consecutive terms was quantified precisely, according to the parity of n. In addition, one incorrect recurrence relation was deleted.

 

 

 

73.

A048909

9-gonal (or nonagonal) triangular numbers. Added a third order homogeneous recurrence, two closed forms and Mathematica code. Also, the limiting value of the ratio of consecutive terms was identified.

 

 

 

74.

A006522

4-dimensional analogue of centred polygonal numbers. Also number of regions created by sides and diagonals of n-gon. For n>=2, terms identified as the diagonal sums of the polygonal numbers listed in increasing rank.

 

 

 

75.

A001107

Decagonal numbers. Second order, inhomogeneous recurrence relation added.

 

 

 

76.

A000567

Octagonal, or star numbers. Second order, inhomogeneous recurrence relation added.

 

 

 

77.

A048908

Indices of triangular numbers which are also 9-gonal. A third order homogeneous recurrence, a closed form and Mathematica code were added and the limiting value of the ratio of consecutive terms was identified.

 

 

 

78.

A048907

Indices of 9-gonal numbers which are also triangular. Added a third order homogeneous recurrence, a closed form and Mathematica code. Also, the limiting value of the ratio of consecutive terms was identified.

 

 

 

79.

A036353

Square pentagonal numbers. Closed form added, together with a third order homogeneous recurrence and Mathematica code. Also, the limiting value of the ratio of consecutive terms was identified and an incorrect generating function deleted.

 

 

 

80.

A046173

Indices of square numbers which are also pentagonal. Closed form added, together with a comment to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(2) + sqrt(3))^4 = 49 + 20 * sqrt(6)”.

 

 

 

81

A046172

Indices of pentagonal numbers which are also square. Closed form added, together with a third order homogeneous recurrence, Mathematica code and a comment to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(2) + sqrt(3))^4 = 49 + 20 * sqrt(6)”.

 

 

 

82.

A046177

Square numbers which are also hexagonal numbers. Added four closed forms, a third order homogeneous recurrence, a generating function and Mathematica code. A comment was also added to the effect that  “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (1 + sqrt(2))^8 = 577 + 408 sqrt(2)”.

 

 

 

83.

A046176

Indices of square numbers which are also hexagonal. A comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (1 + sqrt(2))^4 = 17 + 12 sqrt(2)”. Also, improved Mathematica code given.

 

 

 

84.

A008844

Squares of sequence A001653 : y^2 such that x^2 - 2*y^2 = -1 for some x. Comments added to the effect that the terms of this sequence are the indices of positive hexagonal numbers that are also perfect squares, and also that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (1+sqrt(2))^4 = 17 + 12 sqrt(2)”. In addition two closed forms were added together with a third order homogeneous recurrence, a second order inhomogeneous recurrence and the Mathematica code required to generate the sequence.

 

 

 

85.

A036354

Heptagonal square numbers. Sequence identified as being the union of three subsequences, and two closed forms were given for each. In addition, a seventh order homogeneous recurrence and a sixth order inhomogeneous recurrence were added for the main sequence, together with the Mathematica code required to generate it.

 

 

 

86.

A046196

Indices of square numbers which are also heptagonal. Generating function added, together with the Mathematica code required to generate the sequence.

 

 

 

87.

A046195

Indices of heptagonal numbers (A000566) which are also square. Added the Mathematica code required to generate the sequence, a sixth order inhomogeneous recurrence and an identity relating the members of this sequence to those of  A046196.

 

 

 

88.

A036428

Square octagonal numbers. Four closed forms added together with the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(3))^4 = 97 + 56 sqrt(3)”.

 

 

 

89.

A028230

Bisection of A001353. Indices of square numbers which are also octagonal. Two closed forms added together with the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(3))^2 = 7 + 4 sqrt(3)”. Also, an incorrect recurrence relation was deleted.

 

 

 

90.

A046184

Indices of octagonal numbers which are also square. Four closed forms added together with a second order inhomogeneous recurrence relation, a third order homogeneous recurrence, a generating function and the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(3))^2 = 7 + 4 sqrt(3)”.

 

 

 

91

A036411

9-gonal square numbers. Added  a fourth order inhomogeneous recurrence and the Mathematica code required to generate the sequence. This sequence is actually the union of two subsequences and the limiting values of a(2n)/a(2n-1) and a(2n+1)/a(2n) were also added and quantified.

 

 

 

92.

A048911

Indices of square numbers which are also 9-gonal. Added two closed forms together with a fourth order homogeneous recurrence, a generating function and Mathematica code. In addition, as this sequence is actually the union of two subsequences, the limiting values of the ratios a(2n)/a(2n-1) and a(2n+1)/a(2n) were added and quantified precisely.

 

 

 

93.

A048910

Indices of 9-gonal numbers which are also square. Added two closed forms together with a fourth order inhomogeneous recurrence, a fifth order homogeneous recurrence, a generating function and Mathematica code. In addition, as this sequence is actually the union of two subsequences, the exact forms of the limiting values of the ratios a(2n)/a(2n-1) and a(2n+1)/a(2n) were also added. And finally,  a connection was given between this sequence and A048911(n), whereby

(14 * A048910 (n) – 5) ^ 2 – 56 * A048911(n) ^ 2 = 25 is true for all natural numbers n.

 

 

 

94.

A046180

Hexagonal pentagonal numbers. A third order homogeneous recurrence was added together with four closed forms and the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(3))^8 = 18817 + 10864 sqrt(3)”.

 

 

 

95.

A046178

Indices of pentagonal numbers which are also hexagonal. A third order homogeneous recurrence was added together with two closed forms and the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(3))^4 = 97 + 56 sqrt(3)”.

 

 

 

96.

A046179

Indices of hexagonal numbers which are also pentagonal. A third order homogeneous recurrence was added together with two closed forms and the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(3))^4 = 97 + 56 sqrt(3)”.

 

 

 

97.

A048900

Heptagonal pentagonal numbers. A second order inhomogeneous recurrence and a third order homogeneous recurrence were identified, together with two closed forms, a generating function and the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio

r = lim(n -> Infinity, a(n)/a(n-1)) = (4 + sqrt(15))^4 = 1921 + 496 sqrt(15)”.

 

 

 

98.

A046199

Indices of pentagonal numbers which are also heptagonal. A second order inhomogeneous recurrence and a third order homogeneous recurrence were identified, together with two closed forms, a generating function and the Mathematica code required to generate the sequence. An additional comment was added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (4+sqrt(15))^2 = 31 + 8 sqrt(15)”.

 

 

 

99.

A046198

Indices of heptagonal numbers (A000566) which are also pentagonal. A third order homogeneous recurrence and a second order inhomogeneous recurrence were identified together with two closed forms, a generating function and the Mathematica code required to generate the sequence. A further comment was added that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (4+sqrt(15))^2 = 31 + 8 sqrt(15)”.

 

 

 

100.

A046189

Octagonal pentagonal numbers. A fourth order inhomogeneous recurrence and a fifth order homogeneous recurrence were identified together with two closed forms, a generating function and Mathematica code. This sequence is actually the union of two subsequences and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also evaluated and given.

 

 

 

101.

A046187

Indices of pentagonal numbers which are also octagonal. A fifth order homogeneous recurrence and a fourth order inhomogeneous recurrence were identified together with two closed forms, a generating function and Mathematica code. A046187 is actually the union of two subsequences and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also evaluated and given.

 

 

 

102.

A046188

Indices of octagonal numbers which are also pentagonal. A fifth order homogeneous recurrence and a fourth order inhomogeneous recurrence were identified together with two closed forms, a generating function and Mathematica code.  A046188  is actually the union of two subsequences and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also evaluated and given.

 

 

 

103.

A048915

9-gonal pentagonal numbers. A fifth order homogeneous recurrence and a fourth order inhomogeneous recurrence were identified together with two closed forms, a generating function and Mathematica code. A048915 is actually the union of two subsequences and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also evaluated and given.

 

 

 

104.

A048914

Indices of pentagonal numbers which are also 9-gonal. A fifth order homogeneous recurrence and a fourth order inhomogeneous recurrence were identified together with two closed forms, a generating function and Mathematica code. A048914 is actually the union of two subsequences and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also evaluated and given.

 

 

 

105.

A048913

Indices of 9-gonal numbers which are also pentagonal. A fifth order homogeneous recurrence and a fourth order inhomogeneous recurrence were identified together with two closed forms, a generating function and Mathematica code. A048913  is actually the union of two subsequences and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also evaluated and given.

 

 

 

106.

A048903

Heptagonal hexagonal numbers. A third order homogeneous recurrence and a second order inhomogeneous recurrence were identified together with two closed forms, a generating function and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(5))^8 = 51841 + 23184 sqrt(5)”.

 

 

 

107.

A048901

Indices of hexagonal numbers which are also heptagonal. A second order inhomogeneous recurrence was identified together with two closed forms and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2+sqrt(5))^4 = 161 + 72 sqrt(5)”.

 

 

 

108.

A048902

Indices of heptagonal numbers (A000566) which are also hexagonal. A second order inhomogeneous recurrence was identified together with two closed forms and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (2 + sqrt(5))^4 = 161 + 72 sqrt(5)”.

 

 

 

109.

A046192

Octagonal hexagonal numbers. A third order homogeneous recurrence and a second order inhomogeneous recurrence were identified together with two closed forms, a generating function and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(3) + sqrt(2))^8 = 4801 + 1960 sqrt(6)”.

 

 

 

110.

A046191

Indices of hexagonal numbers which are also octagonal. A second order inhomogeneous recurrence was identified together with two closed forms and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(3) + sqrt(2))^4 = 49 + 20 sqrt(6)”.

 

 

 

111.

A046190

Indices of octagonal numbers which are also hexagonal numbers. A second order inhomogeneous recurrence was identified together with two closed forms and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(3) + sqrt(2))^4 = 49 + 20 sqrt(6)”.

 

 

 

112.

A048918

9-gonal hexagonal numbers. A fifth order homogeneous recurrence and a fourth order inhomogeneous recurrence were specified together with two closed forms, a generating function and Mathematica code. A048918  is actually the union of two subsequences, and the limiting values of  a(2n+1)/a(2n) and a(2n)/a(2n-1) were also identified as forming an approximate 2-cycle whose upper and lower bounds were also evaluated and given.

 

 

 

113.

A048917

Indices of hexagonal numbers which are also 9-gonal. A fourth order inhomogeneous recurrence and two closed forms were given, together with a comment to the effect that  “as n increases, the ratio of consecutive terms settles into an approximate 2-cycle with the ratio a(n)/a(n-1) bounded above and below by 2024 + 765*sqrt(7) and 8 + 3*sqrt(7) respectively”.

 

 

 

114.

A048916

Indices of 9-gonal numbers which are also hexagonal. A fourth order inhomogeneous recurrence and two closed forms were given, together with a comment to the effect that  “as n increases, the ratio of consecutive terms settles into an approximate 2-cycle with the ratio a(n)/a(n-1) bounded above and below by 2024 + 765*sqrt(7) and 8 + 3*sqrt(7) respectively”.

 

 

 

115.

A048906

Octagonal heptagonal numbers. A second order inhomogeneous recurrence was identified along with two closed forms. A comment was also added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(5)+sqrt(6))^8 = 116161 + 21208 sqrt(30)”.

 

 

 

116.

A048904

Indices of heptagonal numbers (A000566) which are also octagonal. Two closed forms were identified together with a second order inhomogeneous recurrence. A comment was also added to the effect that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (sqrt(5)+sqrt(6))^4 = 241 + 44 sqrt(30)”.

 

 

 

117.

A048905

Indices of octagonal numbers which are also heptagonal. Two closed forms were identified along with a second order inhomogeneous recurrence. A comment was also added to the effect that this sequence is approximately geometric with r = (sqrt(5)+sqrt(6))^4 = 241 + 44 sqrt(30)”.

 

 

 

118.

A048921

9-gonal heptagonal numbers (A000566). A third order homogeneous recurrence and a second order inhomogeneous recurrence were identified together with two closed forms, a generating function and the Mathematica code required to generate the sequence. A further comment was added to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (6 + sqrt(35))^4 = 10081 + 1704 sqrt(35)”.

 

 

 

119.

A048920

Indices of heptagonal numbers (A000566) which are also 9-gonal.  Added a second order inhomogeneous recurrence together with a closed form and a comment to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (6 + sqrt(35))^2 = 71 + 12 sqrt(35)”.

 

 

 

120.

A048919

Indices of 9-gonal numbers which are also heptagonal. Added a second order inhomogeneous recurrence together with a closed form and a comment to the effect that that “As n increases, this sequence is approximately geometric with common ratio r = lim(n -> Infinity, a(n)/a(n-1)) = (6 + sqrt(35))^2 = 71 + 12 sqrt(35)”.

 

 

 

121.

A048924

9-gonal octagonal numbers. Two closed forms were identified, together with a second order inhomogeneous recurrence and a generating function. A comment was also added that this sequence is approximately geometric with r = (sqrt(6)+sqrt(7))^8 = 227137 + 35048 sqrt(42)”.

 

 

 

122.

A048923

Indices of octagonal numbers which are also 9-gonal. Two closed forms were identified along with a second order inhomogeneous recurrence. A comment was also added to the effect that this sequence is approximately geometric with r = (sqrt(6)+sqrt(7))^4 = 337 + 52 sqrt(42)”.

 

 

 

123.

A048922

Indices of 9-gonal numbers which are also octagonal. Two closed forms were identified along with a second order inhomogeneous recurrence. A comment was also added to the effect that this sequence is approximately geometric with r = (sqrt(6)+sqrt(7))^4 = 337 + 52 sqrt(42)”.

 

 

 

124.

A061205

a(n) = n times R(n) where R(n) (A004086) is the digit reversal of n. Mathematica code added.

 

 

 

125.

A010690

Period 2: repeat (1,9). Mathematica code added together with a comment to the effect that the terms of this sequence are also the digital roots of those natural numbers that are simultaneously square and triangular.

 

 

 

126.

A188896

Numbers n such that there is no square n-gonal number greater than 1. Mathematica code added.

 

 

 

127.

A146325

Period 3 sequence (1, 4, 1). Comments added to the effect that the terms of this sequence are the digital roots of the centered triangular numbers (A05448) and are connected by the second order inhomogeneous recurrence a(n)=6-a(n-1)-a(n-2).

 

 

 

128.

A005448

Centered triangular numbers: 3n(n-1)/2 + 1. Second order inhomogeneous recurrence relation added, together with a comment that the limiting value of the partial sums of the reciprocals of the terms of this sequence is 2pi/sqrt(15)*tanh(pi/2*sqrt(5/3)).

 

 

 

129.

A006003

n * (n^2 + 1) / 2 (aka partial sums of the centered triangular numbers). Third order inhomogeneous recurrence relation added.

 

 

 

130.

A001844

Centered square numbers: 2n(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; then sequence gives Z values. Second order inhomogeneous recurrence relation added, together with a comment that the limiting value of the partial sums of the reciprocals of the terms of this sequence is pi/2*tanh(pi/2). Three further comments were also added, one stating that all the members of this sequence are congruent to 1 (mod4) and the other two identifying the periodic properties of the sequences of digital roots and units’ digits.

 

 

 

131.

A005891

Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net. Second order inhomogeneous recurrence relation added with a comment that the limit of the partial sums of the reciprocals of the terms of this sequence is 2*pi/sqrt(15)*tanh(pi/2*sqrt(3/5)). Three further comments were also added, one stating that all the terms of this sequence are congruent to 1 (mod5) and the other two identifying the periodic properties of the sequences of digital roots and units’ digits.

 

 

 

132.

A003215

Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice). Comment added that the limit of the partial sums of the reciprocals of the terms of this sequence is pi/sqrt(3)*tanh(pi/(2*sqrt(3))), together with another comment stating that the digital roots of the a(n) form a purely periodic palindromic 3-cycle <1, 7, 1>.

 

 

 

133.

A069099

Centered heptagonal numbers. Second order inhomogeneous recurrence relation added with a comment that the limit of the partial sums of the reciprocals of the terms of this sequence is 2*Pi/sqrt(7)*tanh(Pi/(2*sqrt(7))) = 1.264723171685652.... Three further comments were also added, one stating that all the terms of this sequence are congruent to 1 (mod7) and the other two identifying the periodic properties of the sequences of digital roots and units’ digits.

 

 

 

134.

A060544

Centered 9-gonal (or nonagonal) numbers. Every third triangular number, starting with a(1)=1. Second and third order recurrence relations added, together with a comment that the limiting value of the partial sums of the reciprocals of the terms of this sequence is 2*pi/(3*sqrt(3)). Four further comments were also added, one stating that all the members of this sequence are congruent to 1 (mod9), two identifying the periodic properties of the sequences of digital roots and units’ digits and one showing how to partition the members of this sequence into a sum of 9 triangular numbers.

 

 

 

135.

A015222

Even square pyramidal numbers. Added a sixth order inhomogeneous recurrence relation and a closed form.

 

 

 

136.

A015221

Odd square pyramidal numbers. Added a seventh order homogeneous recurrence, a sixth order inhomogeneous recurrence, a closed form, a generating function and the Mathematica code required to generate the sequence.

 

 

 

137.

A000330

Square pyramidal numbers: 0^2 + 1^2 + 2^2 +...+ n^2 = n*(n+1)*(2*n+1)/6. Added a third order inhomogeneous recurrence together with two comments regarding the periodic behaviour of the sequences of digital roots and units’ digits.

 

 

 

138.

A000292

Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.  Added a third order inhomogeneous recurrence together with a comment regarding the periodic behaviour of the sequence of digital roots.

 

 

 

139.

A193108

The tetrahedral numbers A000292 mod 10. Added a nineteenth order inhomogeneous recurrence.

 

 

 

140.

A015219

Odd tetrahedral numbers. Added a fourth order homogeneous recurrence, a third order inhomogeneous recurrence and the Mathematica code required to generate the sequence.

 

 

 

141.

A004772

Numbers that are not congruent to 1 mod 4. Added a third order inhomogeneous recurrence, a closed form which was constructed using entirely elementary functions and the Mathematica code required to generate the sequence.

 

 

 

142.

A015220

Even tetrahedral numbers. Comment added to the effect that the limiting value of the partial sums of the reciprocals of the even tetrahedral numbers is 3/2 (1-log(2)). Also added a tenth order homogeneous recurrence, a ninth order inhomogeneous recurrence and the Mathematica code required to generate the sequence.

 

 

 

143.

A002411

Pentagonal pyramidal numbers: n^2*(n+1)/2. Added a third order inhomogeneous recurrence, five identities relating the terms of this sequence to the lower ranked pyramidal and figurate polygonal numbers and the binomial coefficients, and two comments regarding the periodic behaviour of the sequences of digital roots and units’ digits.

 

 

 

144.

A015223

Odd pentagonal pyramidal numbers. Added a fourth order homogeneous recurrence, a third order inhomogeneous recurrence and an exact expression for the limiting value of the partial sums of the reciprocals – specifically (8C-2*pi+pi^2-4*log(2))/8, where C is Catalan’s constant (A006752).

 

 

 

145.

A015224

Even pentagonal pyramidal numbers. Added a tenth order homogeneous recurrence, a ninth order inhomogeneous recurrence, a generating function, Mathematica code and an exact expression for the limiting value of the partial sums of the reciprocals – specifically log(2)/2 + pi/4 + 5*pi^2/24 – 2 – C = 0.27217…,, where C is Catalan’s constant (A006752).

 

 

 

146.

A002412

Hexagonal pyramidal numbers, or greengrocer's numbers.  Added a third order inhomogeneous recurrence, a first order non-linear inhomogeneous recurrence, six identities relating the terms of this sequence to the binomial coefficients and lower ranked pyramidal and polygonal numbers, and an exact expression for the limiting value of the partial sums of the reciprocals. Two further comments were also included regarding the periodic behaviour of the sequences of digital roots and units’ digits.

 

 

 

147.

A015225

Odd hexagonal pyramidal numbers. Added a seventh order homogeneous recurrence, a sixth order inhomogeneous recurrence, a closed form, a generating function and the Mathematica code required to generate the sequence.

 

 

 

148.

A015226

Even hexagonal pyramidal numbers. Added a seventh order homogeneous recurrence, a sixth order inhomogeneous recurrence, a closed form, a generating function and different Mathematica code.

 

 

 

149.

A002413

Heptagonal pyramidal numbers.  Added a third order inhomogeneous recurrence, a first order non-linear inhomogeneous recurrence, a fourth order inhomogeneous recurrence, 7 identities relating the terms of this sequence to the binomial coefficients and lower ranked pyramidal and polygonal numbers, and an exact expression for the limiting value of the partial sums of the reciprocals. Two further comments were also included regarding the periodic behaviour of the sequences of digital roots and units’ digits, together with the Mathematica code required to generate the sequence.

 

 

 

150.

A002414

Octagonal pyramidal numbers: n(n+1)(2n-1)/2.  Added a third order inhomogeneous recurrence, a first order non-linear inhomogeneous recurrence, a fourth order inhomogeneous recurrence, seven identities relating the terms of this sequence to the binomial coefficients and lower ranked pyramidal and polygonal numbers, and an exact expression for the limiting value of the partial sums of the reciprocals. Three further comments were also included regarding the structure of the sequence and its periodic behaviour when reduced modulo m.

 

 

 

151.

A007854

9-gonal (or enneagonal) pyramidal numbers: n(n+1)(7n-4)/6. Added a third order inhomogeneous recurrence, a first order non-linear inhomogeneous recurrence, a fourth order inhomogeneous recurrence and five identities relating the terms of this sequence to the binomial coefficients and lower ranked pyramidal and polygonal numbers. Two further comments were also added regarding the periodic behaviour of the sequences of digital roots and units’ digits, together with the Mathematica code required to generate the sequence.

 

 

 

152.

A007585

10-gonal (or decagonal) pyramidal numbers: n(n+1)(8n-5)/6.  Added a third order inhomogeneous recurrence, a first order non-linear inhomogeneous recurrence, a fourth order inhomogeneous recurrence and six identities relating the terms of this sequence to the binomial coefficients and lower ranked pyramidal and polygonal numbers. Further comments were also added regarding the periodic behaviour of the sequences of digital roots and units’ digits, together with the fact that the members of this sequence are the partial sums of the decagonal numbers A001107(n).

 

 

 

153.

A050456

Sum_{d|n, d=1 mod 4} d^4 - Sum_{d|n, d=3 mod 4} d^4. Mathematica code added.

 

 

 

154.

A050468

Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4. Mathematica code added.

 

 

 

155.

A030212

Expansion of eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 in powers of q. Mathematica code added.

 

 

 

156.

A067698

Numbers with relatively many and large divisors (see comments). Improved Mathematica code added and the existing code corrected.

 

 

 

157.

A062089

Decimal expansion of Sierpinski's constant. Mathematica code added.

 

 

 

158.

A004018

Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Comment added to the effect that the zeros in this sequence correspond to those integers with an equal number of 4k+1 and 4k+3 divisors, or equivalently to those that have at least one 4k+3 prime factor with an odd exponent.

 

 

 

159.

A014198

Number of integer solutions to x^2+y^2 <= n excluding (0,0). Mathematica code added, together with a comment to the effect that the terms of this sequence are four times the running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors.

 

 

 

160.

A014200

Number of solutions to x^2+y^2 <= n, excluding (0,0), divided by 4. Mathematica code added, together with a comment to the effect that the terms of this sequence represent the running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors. A goodness of fit of the approximation a(n)~n*pi/4 was also given for n=10^6.

 

 

 

161.

A164940

Partial sums of A138202. Mathematica code added together with a comment to the effect that the limiting value of a(n)/n^2 is 8*pi^4 / (21 zeta(3)).

 

 

 

162.

A000132

Number of ways of writing n as a sum of 5 squares. Comment added to the effect that the units’ digits of a(n) are 2 if n=5t^2 for some natural number t, and 0 otherwise, together with a reference to Moreno and Wagstaff.

 

 

 

162.

A056911

Odd squarefree numbers. Mathematica code added.

 

 

 

163.

A138202

a(n) = A005875(n)^2. Mathematica code added.

 

 

 

164.

A000578

The cubes: a(n) = n^3. Added a third order inhomogeneous recurrence.

 

 

 

165.

A003325

Numbers that are the sum of 2 positive cubes. Comment added to the effect that “This is an infinite sequence in which the first term is prime but thereafter all terms are composite.”

 

 

 

166.

A035504

Numbers that eventually reach 1 under "x -> sum of cubes of digits of x". Mathematica code added.

 

 

 

167.

A165333

Numbers that eventually reach 370 under "x -> sum of cubes of digits of x". Mathematica code added.

 

 

 

168.

A165334

Numbers that eventually reach 371 under "x -> sum of cubes of digits of x". Mathematica code added.

 

 

 

169.

A165335

Numbers that eventually reach 407 under "x -> sum of cubes of digits of x". Mathematica code added.

 

 

 

170.

A030196

Distinct elements occurring in triangle of Eulerian numbers (sorted). Mathematica code added.

 

 

 

171.

A001597

Perfect powers: m^k where m > 0 and k >= 2. Alternative Mathematica code added.

 

 

 

172.

A085323

n is here if it is a sum of two positive cubes, a term from A003325, such that n+1 is also (another) similar sum, i.e. next the term in A003325. Added a comment (with a proof) that this sequence is infinite.

 

 

 

173.

A050985

Cubefree part of n. Comment added regarding the unusual structure of the sequence together with an additional two cross references.

 

 

 

174.

A000583

Fourth powers: a(n) = n^4. Added a fourth order inhomogeneous recurrence.

 

 

 

175.

A000538

Sum of fourth powers: 0^4+1^4+...+n^4. Added a fifth order inhomogeneous recurrence.

 

 

 

176.

A000584

5th powers: a(n) = n^5. Added a fifth order inhomogeneous recurrence.

 

 

 

177

A000539

Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5. Added a sixth order inhomogeneous recurrence.

 

 

 

178.

A001014

6th powers: a(n) = n^6. Signature added for a seventh order homogeneous recurrence, together with a sixth order inhomogeneous recurrence.

 

 

 

179.

A000540

Sum of 6th powers: 1^6 + 2^6 + ... + n^6. Added a seventh order inhomogeneous recurrence.

 

 

 

180.

A001015

Seventh powers: a(n) = n^7. Added a seventh order inhomogeneous recurrence.

 

 

 

181.

A000541

Sum of 7th powers: 1^7 + 2^7 + ... + n^7. Added an eighth order inhomogeneous recurrence.

 

 

 

182.

A001016

Eighth powers: a(n) = n^8. Added an eighth order inhomogeneous recurrence.

 

 

 

183.

A000542

Sum of 8th powers: 1^8 + 2^8 + ... + n^8. Added a ninth order inhomogeneous recurrence.

 

 

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