A276159 - OEIS

%I #48 Feb 16 2025 08:33:36

%S 1,16,106,416,1211,2912,6132,11712,20757,34672,55198,84448,124943,

%T 179648,252008,345984,466089,617424,805714,1037344,1319395,1659680,

%U 2066780,2550080,3119805,3787056,4563846,5463136,6498871,7686016,9040592,10579712,12321617,14285712,16492602,18964128

%N Convolution of nonzero octagonal numbers (A000567) with themselves.

%H OEIS Wiki, <a href="http://oeis.org/wiki/Figurate_numbers">Figurate numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OctagonalNumber.html">Octagonal Number</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1)

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

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

%F 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).

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

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 16, 106, 416, 1211, 2912}, 36]

%t Table[(n + 1) (n + 2) (n + 3) ((9 n^2 + 6 n + 5)/30), {n, 0, 35}]

%Y Cf. A000567.

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

%K nonn,easy,changed

%O 0,2

%A _Ilya Gutkovskiy_, Sep 05 2016