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




3. 
Amicable numbers. Mathematica code added. 




4. 
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. 
Digital roots of the triangular numbers.
Considerably simplified closed form added. 




6. 
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. 
Digital roots of the square numbers. Generating
function added. 




8. 
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. 
Numbers that are not the sum of two not necessarily
distinct abundant numbers. Four references to the literature added. 




10. 

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



11. 
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. 
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. 
Numbers that are not the sum of 4 distinct nonzero squares.
Mathematica code added and a thirty fifth order recurrence relation
identified. 




14. 
Numbers that are not the sum of 5 distinct squares.
Mathematica code added. 




15. 
Numbers that are the sum of two squares of nonnegative
integers. Comment added regarding the structure of such numbers and a
further comment added regarding their properties. 




16. 
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. 
Number of partitions of n into sums of two squares (of
nonnegative integers). Mathematica code added and a formula given,
together with a reference to Hirschhorn’s 2000
paper “Some formulae for partitions into squares”. 




18. 
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. 
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. 
Numbers that are the sum of four but no fewer nonzero
squares. Improved Mathematica code added. 




21. 
Number of partitions of n into sums of three squares of
nonnegative 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. 
Numbers that are the sum of four distinct nonzero squares.
Mathematica code added. 




23. 
Number of partitions of n into sums of four squares of
nonnegative 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. 
“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. 
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. 
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. 
Digital roots of triangular numbers – generating
function added for n>0. 




28. 
Units’ digits of triangular numbers – closed form
added 




29. 
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. 
Odd triangular numbers. Fifth order recurrence
relation added, together with a closed form. 




31. 
Congruent to 0 or 3 mod 4. Improved Mathematica code
added. 




32. 
Even triangular numbers. Fifth order recurrence
relation added, together with a closed form and two different Mathematica
routines required to generate the sequence. 




33. 
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. 
Sums of two or more consecutive integers. Comment regarding
polite numbers corrected and improved Mathematica code added. 




35. 
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. 
Numbers which are the sum of two positive triangular
numbers. Mathematica code added. 




37. 
Least number with exactly n odd divisors. Mathematica
code added. 




38. 
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 crossreference to A000217 and a reference to
Ewell’s 1992 paper “On Sums of Triangular Numbers and Sums of Squares”. 




39. 
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. 
Number of (ordered) ways of writing n as the sum of 2
triangular numbers. Mathematica code added. 




41. 
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. 
Sum of divisors of 2n+1. Efficient Mathematica
code added. 




43. 
Fourier coefficients of E_{\infty,4},
where E_{\infty,4} is the unique normalized weight4 modular form for
\Gamma_0(2) with simple zeros at i*\infty. Mathematica code added. 




44. 
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. 
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. 
Numbers n such that n and 2n1 are primes. Misdirected
link removed. 




47. 
Primes such that their double is 1 away from a prime
number. Mathematica code added. 




48. 
a(0) = 




49. 
Triangular numbers whose digit reversal is also a
triangular number. Mathematica code added. 




50. 
Harshad (Niven) triangular numbers:
triangular numbers which are divisible by the sum of their digits. Mathematica
code added. 




51. 
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. 
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. 
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. 
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. 
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. 
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. 
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. 
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. 
Final digit of sum of all the natural numbers from n to
2*n1. 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. 
The largest ndigit number with 3 divisors.
Assistance given to a commentator acknowledged. 




61. 
Odd pentagonal numbers. Closed form and a fourth
order inhomogeneous recurrence relation added. 




62. 
Even pentagonal numbers. Closed form added, together
with a fourth order inhomogeneous recurrence and a fifth order homogeneous
recurrence. 




63. 
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. 
Generalised pentagonal numbers. Fourth order, inhomogeneous
recurrence relation added. Also a comment added later (with justification) to
the effect that this is a selfgenerating sequence that can be simply
constructed from knowledge of the first term alone. 




65. 
(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. 
Regular figurative or polygonal numbers of order greater
than 2. Mathematica code corrected and link to content of A090428
noted. 




67. 
(9^n1)/8. An existing comment made precise,
improved Mathematica code added together with a second order homogeneous
recurrence relation. 




68. 
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)^(n1)" or
equivalently of any term of A054753^(n1). 




69. 
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. 
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. 
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. 
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. 
9gonal (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. 
4dimensional analogue of centred polygonal numbers.
Also number of regions created by sides and diagonals of ngon. For n>=2, terms identified as the diagonal sums
of the polygonal numbers listed in increasing rank. 




75. 
Decagonal numbers. Second order, inhomogeneous
recurrence relation added. 




76. 
Octagonal, or star numbers. Second order, inhomogeneous
recurrence relation added. 




77. 
Indices of triangular numbers which are also 9gonal.
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. 
Indices of 9gonal 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. 
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. 
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(n1)) = (sqrt(2) + sqrt(3))^4 = 49 + 20 *
sqrt(6)”. 




81 
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(n1)) = (sqrt(2) + sqrt(3))^4 = 49 + 20 * sqrt(6)”. 




82. 
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(n1)) = (1 + sqrt(2))^8 = 577 + 408 sqrt(2)”. 




83. 
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(n1)) = (1 + sqrt(2))^4 = 17 + 12 sqrt(2)”. Also, improved Mathematica
code given. 




84. 
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(n1)) = (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. 
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. 
Indices of square numbers which are also heptagonal.
Generating function added, together with the Mathematica code required to
generate the sequence. 




87. 
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. 
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(n1)) = (2 + sqrt(3))^4 = 97 + 56 sqrt(3)”. 




89. 
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(n1)) = (2 + sqrt(3))^2 = 7 + 4 sqrt(3)”. Also,
an incorrect recurrence relation was deleted. 




90. 
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(n1)) = (2 +
sqrt(3))^2 = 7 + 4 sqrt(3)”. 




91 
9gonal 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(2n1) and a(2n+1)/a(2n) were also added and quantified. 




92. 
Indices of square numbers which are also 9gonal.
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(2n1) and a(2n+1)/a(2n) were added and quantified precisely. 




93. 
Indices of 9gonal 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(2n1) 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. 
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(n1)) = (2 +
sqrt(3))^8 = 18817 + 10864 sqrt(3)”. 




95. 
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(n1)) = (2 + sqrt(3))^4 = 97 + 56 sqrt(3)”. 




96. 
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(n1)) = (2 + sqrt(3))^4 = 97 + 56 sqrt(3)”. 




97. 
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(n1)) = (4 + sqrt(15))^4 = 1921 + 496 sqrt(15)”. 




98. 
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(n1)) = (4+sqrt(15))^2 = 31 + 8 sqrt(15)”. 




99. 
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(n1)) = (4+sqrt(15))^2 = 31 + 8 sqrt(15)”. 




100. 
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(2n1) were also evaluated and given. 




101. 
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(2n1) were
also evaluated and given. 




102. 
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(2n1) were also evaluated and given. 




103. 
9gonal 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(2n1) were also evaluated and given. 




104. 
Indices of pentagonal numbers which are also 9gonal.
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(2n1) were also evaluated and given. 




105. 
Indices of 9gonal 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(2n1) were also evaluated and given. 




106. 
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(n1)) = (2 + sqrt(5))^8 = 51841 + 23184
sqrt(5)”. 




107. 
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(n1)) = (2+sqrt(5))^4 = 161 + 72 sqrt(5)”. 




108. 
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(n1)) = (2 + sqrt(5))^4 = 161 + 72 sqrt(5)”. 




109. 
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(n1)) = (sqrt(3) + sqrt(2))^8 = 4801 + 1960
sqrt(6)”. 




110. 
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(n1)) = (sqrt(3) + sqrt(2))^4 = 49 + 20 sqrt(6)”. 




111. 
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(n1)) = (sqrt(3) + sqrt(2))^4 = 49 + 20
sqrt(6)”. 




112. 
9gonal 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(2n1) were also identified as forming an
approximate 2cycle whose upper and lower bounds were also evaluated and given. 




113. 
Indices of hexagonal numbers which are also 9gonal. 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 2cycle with the ratio a(n)/a(n1) bounded above and below by
2024 + 765*sqrt(7) and 8 + 3*sqrt(7) respectively”. 




114. 
Indices of 9gonal 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 2cycle with the ratio
a(n)/a(n1) bounded above and below by 2024 + 765*sqrt(7) and 8 + 3*sqrt(7)
respectively”. 




115. 
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(n1)) = (sqrt(5)+sqrt(6))^8 =
116161 + 21208 sqrt(30)”. 




116. 
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(n1)) = (sqrt(5)+sqrt(6))^4 = 241 + 44 sqrt(30)”. 




117. 
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. 
9gonal 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(n1)) = (6 +
sqrt(35))^4 = 10081 + 1704 sqrt(35)”. 




119. 
Indices of heptagonal numbers (A000566)
which are also
9gonal. 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(n1)) = (6 + sqrt(35))^2 = 71 + 12 sqrt(35)”. 




120. 
Indices of 9gonal 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(n1)) = (6 + sqrt(35))^2 = 71 + 12 sqrt(35)”. 




121. 
9gonal 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. 
Indices of octagonal numbers which are also 9gonal. 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. 
Indices of 9gonal 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. 
a(n) = n times R(n) where R(n) (A004086)
is the digit
reversal of n. Mathematica
code added. 




125. 
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. 
Numbers n such that there is no square ngonal number
greater than 1. Mathematica code added. 




127. 
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)=6a(n1)a(n2). 




128. 
Centered triangular numbers: 3n(n1)/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. 
n * (n^2 + 1) / 2 (aka partial sums of the centered
triangular numbers). Third order inhomogeneous recurrence relation
added. 




130. 
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. 
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. 
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 3cycle
<1, 7, 1>. 




133. 
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. 
Centered 9gonal (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. 
Even square pyramidal numbers. Added a sixth order
inhomogeneous recurrence relation and a closed form. 




136. 
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. 
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. 
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. 
The tetrahedral numbers A000292
mod 10. Added a nineteenth order inhomogeneous recurrence. 




140. 
Odd tetrahedral numbers. Added a fourth order homogeneous recurrence, a third order
inhomogeneous recurrence and the Mathematica code required to generate the
sequence. 




141. 
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. 
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 (1log(2)). Also added a tenth order
homogeneous recurrence, a ninth order inhomogeneous recurrence and the
Mathematica code required to generate the sequence. 




143. 
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. 
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 (8C2*pi+pi^24*log(2))/8, where C is Catalan’s
constant (A006752). 




145. 
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. 
Hexagonal pyramidal numbers, or greengrocer's numbers. Added a third order inhomogeneous
recurrence, a first order nonlinear 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. 
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. 
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. 
Heptagonal pyramidal numbers. Added a third order inhomogeneous
recurrence, a first order nonlinear 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. 
Octagonal pyramidal numbers: n(n+1)(2n1)/2. Added a third order inhomogeneous
recurrence, a first order nonlinear 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. 
9gonal (or enneagonal) pyramidal
numbers: n(n+1)(7n4)/6. Added a third order inhomogeneous recurrence,
a first order nonlinear 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. 
10gonal (or decagonal) pyramidal numbers: n(n+1)(8n5)/6. Added a third order inhomogeneous
recurrence, a first order nonlinear 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. 
Sum_{dn, d=1 mod 4} d^4  Sum_{dn, d=3 mod 4} d^4. Mathematica code added. 




154. 
Sum_{dn, n/d=1 mod 4} d^4 
Sum_{dn, n/d=3 mod 4} d^4. Mathematica code
added. 




155. 
Expansion of eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 in
powers of q. Mathematica code added. 




156. 
Numbers with relatively many and large divisors (see
comments). Improved
Mathematica code added and the existing code corrected. 




157. 
Decimal expansion of Sierpinski's constant.
Mathematica code added. 




158. 
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. 
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. 
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. 
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. 
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 




162. 
Odd squarefree numbers. Mathematica code added. 




163. 
a(n) = A005875(n)^2.
Mathematica code added. 




164. 
The cubes: a(n) = n^3. Added a third order inhomogeneous recurrence. 




165. 
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. 
Numbers that eventually reach 1 under "x > sum of
cubes of digits of x". Mathematica
code added. 




167. 
Numbers that eventually reach 370 under "x > sum
of cubes of digits of x". Mathematica code added. 




168. 
Numbers that eventually reach 371 under "x > sum
of cubes of digits of x". Mathematica code added. 




169. 
Numbers that eventually reach 407 under "x > sum
of cubes of digits of x". Mathematica code added. 




170. 
Distinct elements occurring in triangle of Eulerian numbers
(sorted). Mathematica code
added. 




171. 
Perfect powers: m^k where m >
0 and k >= 2. Alternative
Mathematica code added. 




172. 
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. 
Cubefree part of n. Comment added regarding the unusual
structure of the sequence together with an additional two cross references. 




174. 
Fourth powers: a(n) = n^4. Added
a fourth order inhomogeneous recurrence. 




175. 
Sum of fourth powers: 0^4+1^4+...+n^4. Added a fifth
order inhomogeneous recurrence. 




176. 
5th powers: a(n) = n^5. Added
a fifth order inhomogeneous recurrence. 




177 
Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5. Added a sixth order inhomogeneous
recurrence. 




178. 
6th powers: a(n) = n^6. Signature
added for a seventh order homogeneous recurrence, together with a sixth order
inhomogeneous recurrence. 




179. 
Sum of 6th powers: 1^6 + 2^6 + ... + n^6. Added a seventh
order inhomogeneous recurrence. 




180. 
Seventh powers: a(n) = n^7. Added
a seventh order inhomogeneous recurrence. 




181. 
Sum of 7th powers: 1^7 + 2^7 + ... + n^7. Added an eighth
order inhomogeneous recurrence. 




182. 
Eighth powers: a(n) = n^8. Added
an eighth order inhomogeneous recurrence. 




183. 
Sum of 8th powers: 1^8 + 2^8 + ... + n^8. Added a ninth
order inhomogeneous recurrence. 


