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%I #31 Mar 01 2022 05:33:41
%S 0,1,20,23,61,66,123,130,206,215,310,321,435,448,581,596,748,765,936,
%T 955,1145,1166,1375,1398,1626,1651,1898,1925,2191,2220,2505,2536,2840,
%U 2873,3196,3231,3573,3610,3971,4010,4390,4431,4830,4873,5291,5336,5773,5820,6276,6325,6800,6851,7345,7398,7911,7966
%N Generalized 23-gonal (or icositrigonal) numbers: m*(21*m - 19)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...
%C 168*a(n) + 361 is a square. - _Bruno Berselli_, Jul 10 2018
%C Partial sums of A317319. - _Omar E. Pol_, Jul 28 2018
%H Colin Barker, <a href="/A303303/b303303.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 _Colin Barker_, Jun 27 2018: (Start)
%F G.f.: x*(1 + 19*x + x^2) / ((1 - x)^3*(1 + x)^2).
%F a(n) = n*(21*n + 38) / 8 for n even.
%F a(n) = (21*n - 17)*(n + 1) / 8 for n odd.
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F (End)
%F Sum_{n>=1} 1/a(n) = 42/361 + 2*Pi*cot(2*Pi/21)/19. - _Amiram Eldar_, Mar 01 2022
%t CoefficientList[ Series[-x (x^2 + 19x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 20, 23, 61}, 51] (* _Robert G. Wilson v_, Jul 28 2018 *)
%o (PARI) concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ _Colin Barker_, Jun 27 2018
%Y Cf. A051875, A317319.
%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), this sequence (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 _Omar E. Pol_, Jun 24 2018
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