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%I #43 Mar 01 2022 05:33:17
%S 0,1,26,29,79,84,159,166,266,275,400,411,561,574,749,764,964,981,1206,
%T 1225,1475,1496,1771,1794,2094,2119,2444,2471,2821,2850,3225,3256,
%U 3656,3689,4114,4149,4599,4636,5111,5150,5650,5691,6216,6259,6809,6854,7429,7476,8076
%N Generalized 29-gonal (or icosienneagonal) numbers: m*(27*m - 25)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...
%C Numbers k such that 216*k + 625 is a square. - _Bruno Berselli_, Jun 08 2018
%C Partial sums of A317325.
%H Colin Barker, <a href="/A303815/b303815.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 From _Bruno Berselli_, Jun 07 2018: (Start)
%F G.f.: x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3).
%F a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F a(n) = (54*n*(n + 1) + 23*(2*n + 1)*(-1)^n - 23)/16. Therefore:
%F a(n) = n*(27*n + 50)/8, if n is even, or (n + 1)*(27*n - 23)/8 otherwise.
%F 2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(27*n^2 - 25) = 0. (End)
%F Sum_{n>=1} 1/a(n) = 2*(27 + 25*Pi*cot(2*Pi/27))/625. - _Amiram Eldar_, Mar 01 2022
%t Table[(54 n (n + 1) + 23 (2 n + 1) (-1)^n - 23)/16, {n, 0, 50}] (* _Bruno Berselli_, Jun 07 2018 *)
%t CoefficientList[ Series[-x (x^2 + 25x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 26, 29, 79, 84}, 50] (* _Robert G. Wilson v_, Jul 28 2018 *)
%t With[{nn=25},Riffle[Table[1-(29x)/2+(27x^2)/2,{x,nn}],PolygonalNumber[ 29,Range[ nn]]]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Nov 26 2020 *)
%o (PARI) concat(0, Vec(x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ _Colin Barker_, Jun 12 2018
%Y Cf. A255187, A277990 (see the third comment), A316672, A317325.
%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), A277082 (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), this sequence (k=29), A316729 (k=30).
%K nonn,easy
%O 0,3
%A _Omar E. Pol_, Jun 06 2018
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