A303299 Generalized 22-gonal (or icosidigonal) numbers: m*(10*m - 9) with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 19, 22, 58, 63, 117, 124, 196, 205, 295, 306, 414, 427, 553, 568, 712, 729, 891, 910, 1090, 1111, 1309, 1332, 1548, 1573, 1807, 1834, 2086, 2115, 2385, 2416, 2704, 2737, 3043, 3078, 3402, 3439, 3781, 3820, 4180, 4221, 4599, 4642, 5038, 5083, 5497, 5544, 5976, 6025, 6475, 6526, 6994, 7047, 7533, 7588

Offset: 0

Author

Omar E. Pol, Jun 23 2018

Keywords

• nonn
• easy

Comments

Partial sums of A317318. - Omar E. Pol, Jul 28 2018

Exponents in expansion of Product_{n >= 1} (1 + x^(20*n-19))*(1 + x^(20*n-1))*(1 - x^(20*n)) = 1 + x + x^19 + x^22 + x^58 + .... - Peter Bala, Dec 10 2020

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. A051874, A317318.

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), this sequence (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Programs

Maple

a:= n-> (m-> m*(10*m-9))(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 23 2018

Mathematica

CoefficientList[ Series[-x (x^2 + 18x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 19, 22, 58}, 51] (* Robert G. Wilson v, Jul 28 2018 *)
nn=30; Sort[Table[n (10 n - 9), {n, -nn, nn}]] (* Vincenzo Librandi, Jul 29 2018 *)

PARI

a(n) = n++; my(m = (-1) ^ n * (n >> 1)); m * (10 * m - 9) \\ David A. Corneth, Jun 23 2018

PARI

concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jun 23 2018

Formula

From Colin Barker, Jun 23 2018: (Start)

G.f.: x*(1 + 18*x + x^2) / ((1 - x)^3*(1 + x)^2).

a(n) = (5*n^2 + 9*n)/2 for n even.

a(n) = (5*n^2 + n - 4)/2 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) = (10 + 9*sqrt(5+2*sqrt(5))*Pi)/81. - Amiram Eldar, Mar 01 2022