|
%I #38 Mar 01 2022 05:33:25
%S 0,1,21,24,64,69,129,136,216,225,325,336,456,469,609,624,784,801,981,
%T 1000,1200,1221,1441,1464,1704,1729,1989,2016,2296,2325,2625,2656,
%U 2976,3009,3349,3384,3744,3781,4161,4200,4600,4641,5061,5104,5544,5589,6049,6096,6576,6625
%N Generalized 24-gonal (or icositetragonal) numbers: m*(11*m - 10) with m = 0, +1, -1, +2, -2, +3, -3, ...
%C a(25) = 1729 is the Hardy-Ramanujan number.
%C Numbers k such that 11*k + 25 is a square. - _Bruno Berselli_, Jun 08 2018
%C Partial sums of A317320. - _Omar E. Pol_, Jul 28 2018
%H Colin Barker, <a href="/A303814/b303814.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 08 2018: (Start)
%F G.f.: x*(1 + 20*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) = (22*n*(n + 1) + 9*(2*n + 1)*(-1)^n - 9)/8. Therefore:
%F a(n) = n*(11*n + 20)/4, if n is even, or (n + 1)*(11*n - 9)/4 otherwise.
%F (2*n - 1)*a(n) + (2*n + 1)*a(n-1) - n*(11*n^2 - 10) = 0. (End)
%F Sum_{n>=1} 1/a(n) = (11 + 10*Pi*cot(Pi/11))/100. - _Amiram Eldar_, Mar 01 2022
%t With[{pp = 24, nn = 55}, {0}~Join~Riffle[Array[PolygonalNumber[pp, #] &, Ceiling[nn/2]], Array[PolygonalNumber[pp, -#] &, Ceiling[nn/2]]]] (* _Michael De Vlieger_, Jun 06 2018 *)
%t Table[(22 n (n + 1) + 9 (2 n + 1) (-1)^n - 9)/8, {n, 0, 50}] (* _Bruno Berselli_, Jun 08 2018 *)
%t CoefficientList[ Series[-x (x^2 + 20x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 21, 24, 64}, 50] (* _Robert G. Wilson v_, Jul 28 2018 *)
%o (PARI) concat(0, Vec(x*(1 + 20*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ _Colin Barker_, Jun 12 2018
%Y Cf. A051876, A317320.
%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), this sequence (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 06 2018
|
