A344410 a(n) = (3*n^2 - 1) * (3*n^2 - 2) * (3*n^3 - 3*n + 1)/2.

1, 1, 1045, 23725, 195661, 975061, 3578401, 10680265, 27453385, 63016921, 132361021, 258815701, 477132085, 837244045, 1408778281, 2286380881, 3595928401, 5501691505, 8214519205, 12001111741, 17194450141, 24205450501, 33535911025, 45792819865, 61704091801

Offset

0,3

Comments

a(n) is both (3*n+2)-gonal number and (3*n+2)-gonal pyramidal number.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Numberphile, Cannon Ball Numbers.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

Let p(k,m) = A057145(k,m) denote m-th k-gonal number. Then

a(n) = p(3*n+2, 3*n^3-3*n+1);

a(n) = Sum_{j=1..3*n^2-2} p(3*n+2, j) for n > 0.

G.f.: (1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8.

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Sep 05 2022

Mathematica

Table[PolygonalNumber[3*n + 2, 3*n^3 - 3*n + 1], {n, 0, 24}] (* Amiram Eldar, May 17 2021 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 1, 1045, 23725, 195661, 975061, 3578401, 10680265}, 30] (* Harvey P. Dale, Aug 10 2021 *)

Prog

(PARI) a(n) = (3*n^2-1)*(3*n^2-2)*(3*n^3-3*n+1)/2;
(PARI) p(k, n) = n*((k-2)*n-k+4)/2;
a(n) = p(3*n+2, 3*n^3-3*n+1);
(PARI) my(N=40, x='x+O('x^N)); Vec((1-7*x+1065*x^2+15337*x^3+35135*x^4+15567*x^5+943*x^6-x^7)/(1-x)^8)

Crossrefs

Cf. A016789, A027669, A057145, A344376, A344377.

Sequence in context: A334012 A344376 A023067 * A110597 A258130 A226114

Adjacent sequences: A344407 A344408 A344409 * A344411 A344412 A344413

Keyword

nonn,easy

Author

Seiichi Manyama, May 17 2021

Status

approved