A303815 Generalized 29-gonal (or icosienneagonal) numbers: m*(27*m - 25)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 26, 29, 79, 84, 159, 166, 266, 275, 400, 411, 561, 574, 749, 764, 964, 981, 1206, 1225, 1475, 1496, 1771, 1794, 2094, 2119, 2444, 2471, 2821, 2850, 3225, 3256, 3656, 3689, 4114, 4149, 4599, 4636, 5111, 5150, 5650, 5691, 6216, 6259, 6809, 6854, 7429, 7476, 8076

Offset: 0

Author

Omar E. Pol, Jun 06 2018

Keywords

• nonn
• easy

Comments

Numbers k such that 216*k + 625 is a square. - Bruno Berselli, Jun 08 2018

Partial sums of A317325.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A255187, A277990 (see the third comment), A316672, A317325.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), this sequence (k=29), A316729 (k=30).

Programs

Mathematica

Table[(54 n (n + 1) + 23 (2 n + 1) (-1)^n - 23)/16, {n, 0, 50}] (* Bruno Berselli, Jun 07 2018 *)
CoefficientList[ Series[-x (x^2 + 25x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 26, 29, 79, 84}, 50] (* Robert G. Wilson v, Jul 28 2018 *)
With[{nn=25},Riffle[Table[1-(29x)/2+(27x^2)/2,{x,nn}],PolygonalNumber[ 29,Range[ nn]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 26 2020 *)

PARI

concat(0, Vec(x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018

Formula

From Bruno Berselli, Jun 07 2018: (Start)

G.f.: x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3).

a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

a(n) = (54*n*(n + 1) + 23*(2*n + 1)*(-1)^n - 23)/16. Therefore:

a(n) = n*(27*n + 50)/8, if n is even, or (n + 1)*(27*n - 23)/8 otherwise.

2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(27*n^2 - 25) = 0. (End)

Sum_{n>=1} 1/a(n) = 2*(27 + 25*Pi*cot(2*Pi/27))/625. - Amiram Eldar, Mar 01 2022