cp's OEIS Frontend

Note that in the sequences of generalized k-gonal numbers always a(3) = k. In this case k = 27.

Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, with k >= 5.

A general formula for the generalized k-gonal numbers is given by m*((k-2)*m-k+4)/2, with m = 0, +1, -1, +2, -2, +3, -3, ..., k >= 5.

Partial sums of A317323. - Omar E. Pol, Jul 28 2018

Note that in the sequences of generalized k-gonal numbers always a(3) = k. In this case k = 30.

Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, with k >= 5.

A general formula for the generalized k-gonal numbers is given by m*((k-2)*m-k+4)/2, with m = 0, +1, -1, +2, -2, +3, -3, ..., k >= 5.

Every sequence of generalized k-gonal numbers can be represented as vertices of a rectangular spiral constructed with line segments on the square grid, with k >= 5.

56*a(n) + 169 is a square. - Vincenzo Librandi, Jul 12 2018

Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - Omar E. Pol, Jul 27 2018

Also partial sums of A317326. - Omar E. Pol, Jul 28 2018

Numerators are all 1.

Setting x=1/3 into 1/(3*x)*log((1+x)/(1-x)^2) = Sum_{k>=0} x^k/((2-(-1)^k)*(k+1)),

log(3) = Sum_{k>=0} 1/((2-(-1)^k)*(k+1)*3^k) = Sum_{k>=0} (9/(2k+1)+1/(2k+2))/9^(k+1) is obtained.

It appears that this is also the first differences of the generalized decagonal numbers A074377. - Omar E. Pol, Sep 10 2011

It appears that this is also A005408 and positive terms of A008588 interleaved. - Omar E. Pol, May 28 2012

A bisection of A016742. Sequence arises from reading the line from 0, in the direction 0, 16, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Also, sequence found by reading the line from 0, in the direction 0, 16, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Sep 10 2011

Also, column k lists the partial sums of the column k of A195151. The first differences in row n are always the n-th term of the triangular numbers repeated 0,0,1,1,3,3,6,6,... ([0,0] together with A008805).

Also, for k >= 1, this is a table of generalized polygonal numbers since column k lists the generalized m-gonal numbers, where m = k+4, for example: if k = 1 then m = 5, so the column 1 lists the generalized pentagonal numbers A001318 (see example).

If Y_i (i=1,2,3) are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Concentric hexadecagonal numbers or concentric hexakaidecagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 16, ..., and the same line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Main axis, perpendicular to A033996 in the same spiral.

Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13]. For more information see A210843 and other sequences of this family. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 7 (mod 8). - Peter Bala, Dec 10 2020

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...

In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):

========================================================

. Column k of

. this square

. Generalized Triangle Triangle array A211970

k m m-gonal "A" "B" [row sums of

. numbers triangle "B"

. (if k>=1) with a(0)=1,

. if k >= 0]

========================================================

0 4 A008794 - - A211971

1 5 A001318 A195310 A175003 A000041

2 6 A000217 A195826 A195836 A006950

3 7 A085787 A195827 A195837 A036820

4 8 A001082 A195828 A195838 A195848

5 9 A118277 A195829 A195839 A195849

6 10 A074377 A195830 A195840 A195850

7 11 A195160 A195831 A195841 A195851

8 12 A195162 A195832 A195842 A195852

9 13 A195313 A195833 A195843 A196933

10 14 A195818 A210944 A210954 A210964

...

It appears that column 2 of the square array is A006950.

It appears that column 3 of the square array is A036820.

The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

Via the partial fraction decomposition 1/((4n+3)*(4n+5)) = (1/2) *(1/(4n+3) -1/(4n+5)) we find 2*Sum_{n>=0} (-1)^n/a(n) = 2*Sum_{n>=0} (-1)^n/( (4*n+3)*(4*n+5) ) = 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ... = (1/1 + 1/3 -1/5 -1/7 +1/9 +1/11 -1/13 -1/15 +1/17 +1/19 -- ++ ..)-1 = Sum_{n>=0} (-1)^n/A016813(n) + Sum_{n>=0} (-1)^n/A004767(n) -1 = -1 + Sum_{n>=0} b(n)/n^1 where b(n) = 1, 0, 1, 0, -1, 0, -1, 0 is a sequence with period length 8, one of the Dirichlet L-series modulo 8. The alternating sum becomes -1 +L(m=8,r=4,s=1) = Pi*sqrt(2)/4-1 = A093954 - 1.

Pi = 4 - 8*Sum(1/a(n)) noted by Bronstein-Semendjajew for the variant a(n) = (4n-1)*(4n+1) starting at n=1. - Frank Ellermann, Sep 18 2011

The identity (16*n^2-1)^2 - (64*n^2-8)*(2*n)^2 = 1 can be written as a(n)^2 - A158487(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012

Sequence found by reading the line from 15, in the direction 15, 63,... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

Essentially the least common multiple of 4*n+1 and 4*n-1. - Colin Barker, Feb 11 2017