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
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).
Programs
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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 *) -
PARI
a(n) = (3*n^2-1)*(3*n^2-2)*(3*n^3-3*n+1)/2;
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PARI
p(k, n) = n*((k-2)*n-k+4)/2; a(n) = p(3*n+2, 3*n^3-3*n+1);
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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)
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
Comments