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A277082 Generalized 15-gonal (or pentadecagonal) numbers: n*(13*n - 11)/2, n = 0,+1,-1,+2,-2,+3,-3, ... 1

%I

%S 0,1,12,15,37,42,75,82,126,135,190,201,267,280,357,372,460,477,576,

%T 595,705,726,847,870,1002,1027,1170,1197,1351,1380,1545,1576,1752,

%U 1785,1972,2007,2205,2242,2451,2490,2710,2751,2982,3025,3267,3312,3565,3612,3876,3925,4200,4251,4537,4590,4887,4942

%N Generalized 15-gonal (or pentadecagonal) numbers: n*(13*n - 11)/2, n = 0,+1,-1,+2,-2,+3,-3, ...

%C More generally, the ordinary generating function for the generalized k-gonal numbers is x*(1 + (k - 4)*x + x^2)/((1 - x)^3*(1 + x)^2). A general formula for the generalized k-gonal numbers is given by (k*(2*n^2 + 2*((-1)^n + 1)*n + (-1)^n - 1) - 2*(2*n^2 + 2*(3*(-1)^n + 1)*n + 3*((-1)^n - 1)))/16.

%C For k>4, Sum_{n>=1} 1/a(k,n) = 2*(k-2)/(k-4)^2 + 2*Pi*cot(2*Pi/(k-2))/(k-4). - _Vaclav Kotesovec_, Oct 05 2016

%H G. C. Greubel, <a href="/A277082/b277082.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F G.f.: x*(1 + 11*x + x^2)/((1 - x)^3*(1 + x)^2).

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

%F a(n) = (26*n^2 + 26*n + 9*(-1)^n*(2*n+1) - 9)/16.

%F Sum_{n>=1} 1/a(n) = 26/121 + 2*Pi*cot(2*Pi/13)/11 = 1.3032041594895857... . - _Vaclav Kotesovec_, Oct 05 2016

%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 12, 15, 37}, 56]

%t Table[(26 n^2 + 26 n + 9 (-1)^n (2 n + 1) - 9)/16, {n, 0, 55}]

%o (PARI) concat(0, Vec(x*(1+11*x+x^2)/((1-x)^3*(1+x)^2) + O(x^99))) \\ _Altug Alkan_, Oct 01 2016

%Y Cf. A051867 (15-gonal numbers).

%Y Cf. similar 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).

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Sep 29 2016

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Last modified February 22 15:12 EST 2018. Contains 299454 sequences. (Running on oeis4.)