A276159 Convolution of nonzero octagonal numbers (A000567) with themselves.

1, 16, 106, 416, 1211, 2912, 6132, 11712, 20757, 34672, 55198, 84448, 124943, 179648, 252008, 345984, 466089, 617424, 805714, 1037344, 1319395, 1659680, 2066780, 2550080, 3119805, 3787056, 4563846, 5463136, 6498871, 7686016, 9040592, 10579712, 12321617, 14285712, 16492602, 18964128

Offset: 0

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

• nonn
• easy

Links

• OEIS Wiki, Figurate numbers
• Eric Weisstein's World of Mathematics, Octagonal Number
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Crossrefs

Cf. A000567.

Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.

Programs

Mathematica

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 16, 106, 416, 1211, 2912}, 36]
Table[(n + 1) (n + 2) (n + 3) ((9 n^2 + 6 n + 5)/30), {n, 0, 35}]

Formula

O.g.f.: (1 + 5*x)^2/(1 - x)^6.

E.g.f.: (30 + 450*x + 1125*x^2 + 725*x^3 + 150*x^4 + 9*x^5)*exp(x)/30.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = (n + 1)*(n + 2)*(n + 3)*(9*n^2 + 6*n + 5)/30.