A303303 Generalized 23-gonal (or icositrigonal) numbers: m*(21*m - 19)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 20, 23, 61, 66, 123, 130, 206, 215, 310, 321, 435, 448, 581, 596, 748, 765, 936, 955, 1145, 1166, 1375, 1398, 1626, 1651, 1898, 1925, 2191, 2220, 2505, 2536, 2840, 2873, 3196, 3231, 3573, 3610, 3971, 4010, 4390, 4431, 4830, 4873, 5291, 5336, 5773, 5820, 6276, 6325, 6800, 6851, 7345, 7398, 7911, 7966

Offset

0,3

Comments

168*a(n) + 361 is a square. - Bruno Berselli, Jul 10 2018

Partial sums of A317319. - Omar E. Pol, Jul 28 2018

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).

Formula

From Colin Barker, Jun 27 2018: (Start)

G.f.: x*(1 + 19*x + x^2) / ((1 - x)^3*(1 + x)^2).

a(n) = n*(21*n + 38) / 8 for n even.

a(n) = (21*n - 17)*(n + 1) / 8 for n odd.

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

(End)

Sum_{n>=1} 1/a(n) = 42/361 + 2*Pi*cot(2*Pi/21)/19. - Amiram Eldar, Mar 01 2022

Mathematica

CoefficientList[ Series[-x (x^2 + 19x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 20, 23, 61}, 51] (* Robert G. Wilson v, Jul 28 2018 *)

Prog

(PARI) concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Jun 27 2018

Crossrefs

Cf. A051875, A317319.

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), this sequence (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Sequence in context: A268490 A153717 A358424 * A045563 A013339 A199360

Adjacent sequences: A303300 A303301 A303302 * A303304 A303305 A303306

Keyword

nonn,easy

Author

Omar E. Pol, Jun 24 2018

Status

approved