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%I #44 Jul 30 2018 04:03:33
%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
%C Numbers k for which 104*k + 121 is a square. - _Bruno Berselli_, Jul 10 2018
%C Partial sums of A317311. - _Omar E. Pol_, Jul 28 2018
%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
%o (GAP) a:=[0,1,12,15,37];; for n in [6..60] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]-a[n-4]+a[n-5]; od; a; # _Muniru A Asiru_, Jul 10 2018
%Y Cf. A051867 (15-gonal numbers), A316672, A317311.
%Y 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), this sequence (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Sep 29 2016
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