|
%I #30 Mar 01 2022 05:33:28
%S 0,1,22,25,67,72,135,142,226,235,340,351,477,490,637,652,820,837,1026,
%T 1045,1255,1276,1507,1530,1782,1807,2080,2107,2401,2430,2745,2776,
%U 3112,3145,3502,3537,3915,3952,4351,4390,4810,4851,5292,5335,5797,5842,6325,6372,6876,6925
%N Generalized 25-gonal (or icosipentagonal) numbers: m*(23*m - 21)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...
%C Numbers k for which 184*k + 441 is a square. - _Bruno Berselli_, Jul 10 2018
%C Partial sums of A317321. - _Omar E. Pol_, Jul 28 2018
%H Colin Barker, <a href="/A303304/b303304.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_, Jul 10 2018: (Start)
%F G.f.: x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2).
%F a(n) = n*(23*n + 42)/8 for n even.
%F a(n) = (23*n - 19)*(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) = 46/441 + 2*Pi*cot(2*Pi/23)/21. - _Amiram Eldar_, Mar 01 2022
%p seq(coeff(series(x*(x^2+21*x+1)/((1-x)^3*(1+x)^2), x,n+1),x,n),n=0..50); # _Muniru A Asiru_, Jul 10 2018
%t CoefficientList[Series[x (1 + 21 x + x^2)/((1 - x)^3*(1 + x)^2), {x, 0, 49}], x] (* or *)
%t Array[PolygonalNumber[25, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 50, 0] (* _Michael De Vlieger_, Jul 10 2018 *)
%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 22, 25, 67}, 50] (* _Robert G. Wilson v_, Jul 15 2018 *)
%o (PARI) concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ _Colin Barker_, Jul 10 2018
%o (GAP) a:=[0,1,22,25,67];; for n in [6..50] 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. A255184, A317321.
%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), this sequence (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_, Jul 10 2018
|
