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 A303815 Generalized 29-gonal (or icosienneagonal) numbers: m*(27*m - 25)/2 with m = 0, +1, -1, +2, -2, +3, -3, ... 31
 0, 1, 26, 29, 79, 84, 159, 166, 266, 275, 400, 411, 561, 574, 749, 764, 964, 981, 1206, 1225, 1475, 1496, 1771, 1794, 2094, 2119, 2444, 2471, 2821, 2850, 3225, 3256, 3656, 3689, 4114, 4149, 4599, 4636, 5111, 5150, 5650, 5691, 6216, 6259, 6809, 6854, 7429, 7476, 8076 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Numbers k such that 216*k + 625 is a square. - Bruno Berselli, Jun 08 2018 Partial sums of A317325. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). FORMULA From Bruno Berselli, Jun 07 2018: (Start) G.f.: x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3). a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). a(n) = (54*n*(n + 1) + 23*(2*n + 1)*(-1)^n - 23)/16. Therefore: a(n) = n*(27*n + 50)/8, if n is even, or (n + 1)*(27*n - 23)/8 otherwise. 2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(27*n^2 - 25) = 0. (End) Sum_{n>=1} 1/a(n) =  2*(27 + 25*Pi*cot(2*Pi/27))/625. - Amiram Eldar, Mar 01 2022 MATHEMATICA Table[(54 n (n + 1) + 23 (2 n + 1) (-1)^n - 23)/16, {n, 0, 50}] (* Bruno Berselli, Jun 07 2018 *) CoefficientList[ Series[-x (x^2 + 25x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 26, 29, 79, 84}, 50] (* Robert G. Wilson v, Jul 28 2018 *) 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 *) PROG (PARI) concat(0, Vec(x*(1 + 25*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018 CROSSREFS Cf. A255187, A277990 (see the third comment), A316672, A317325. 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). Sequence in context: A106550 A304949 A316617 * A069962 A178098 A278779 Adjacent sequences:  A303812 A303813 A303814 * A303816 A303817 A303818 KEYWORD nonn,easy AUTHOR Omar E. Pol, Jun 06 2018 STATUS approved

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Last modified July 3 16:27 EDT 2022. Contains 355055 sequences. (Running on oeis4.)