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%I #42 Feb 28 2022 04:13:37
%S 0,1,18,21,55,60,111,118,186,195,280,291,393,406,525,540,676,693,846,
%T 865,1035,1056,1243,1266,1470,1495,1716,1743,1981,2010,2265,2296,2568,
%U 2601,2890,2925,3231,3268,3591,3630,3970,4011,4368,4411,4785,4830,5221,5268,5676,5725,6150,6201,6643,6696,7155,7210
%N Generalized 21-gonal (or icosihenagonal) numbers: m*(19*m - 17)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...
%C Numbers k for which 152*k + 289 is a square. - _Bruno Berselli_, Jul 10 2018
%C Partial sums of A317317. - _Omar E. Pol_, Jul 28 2018
%H Colin Barker, <a href="/A303298/b303298.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 G.f.: -(x^2+17*x+1)*x/((x+1)^2*(x-1)^3). - _Alois P. Heinz_, Jun 23 2018
%F From _Colin Barker_, Jun 24 2018: (Start)
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F a(n) = (19*n^2 + 34*n) / 8 for n even.
%F a(n) = (19*n^2 + 4*n - 15) / 8 for n odd.
%F (End)
%F Sum_{n>=1} 1/a(n) = 38/289 + 2*Pi*cot(2*Pi/19)/17. - _Amiram Eldar_, Feb 28 2022
%p a:= n-> (m-> m*(19*m-17)/2)(-ceil(n/2)*(-1)^n):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Jun 23 2018
%t CoefficientList[Series[-(x^2 + 17 x + 1) x/((x + 1)^2*(x - 1)^3), {x, 0, 55}], x] (* or *)
%t Array[PolygonalNumber[21, (1 - 2 Boole[EvenQ@ #]) Ceiling[#/2]] &, 56, 0] (* _Michael De Vlieger_, Jul 10 2018 *)
%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 18, 21, 55}, 51] (* _Robert G. Wilson v_, Jul 28 2018 *)
%o (PARI) concat(0, Vec(x*(1 + 17*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jun 24 2018
%o (GAP) a:=[0,1,18,21,55];; for n in [6..60] 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. A051873, A316672, A317317.
%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), this sequence (k=21), A303299 (k=22), A303303 (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 23 2018
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