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.	A178543	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.

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