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%I #32 Jul 31 2018 09:56:42
%S 0,1,16,19,49,54,99,106,166,175,250,261,351,364,469,484,604,621,756,
%T 775,925,946,1111,1134,1314,1339,1534,1561,1771,1800,2025,2056,2296,
%U 2329,2584,2619,2889,2926,3211,3250,3550,3591,3906,3949,4279,4324,4669,4716,5076,5125,5500,5551,5941,5994,6399
%N Generalized 19-gonal (or enneadecagonal) numbers: m*(17*m - 15)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...
%C Numbers k for which 136*k + 225 is a square. - _Bruno Berselli_, Jul 10 2018
%C Partial sums of A317315. - _Omar E. Pol_, Jul 28 2018
%H Colin Barker, <a href="/A303813/b303813.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 08 2018: (Start)
%F G.f.: x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2).
%F a(n) = (34*n^2 + 60*n)/16 for n even.
%F a(n) = (34*n^2 + 8*n - 26)/16 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)
%t With[{nn = 54}, {0}~Join~Riffle[Array[PolygonalNumber[19, #] &, Ceiling[nn/2]], Array[PolygonalNumber[19, -#] &, Ceiling[nn/2]]]] (* _Michael De Vlieger_, Jun 06 2018 *)
%t CoefficientList[ Series[-x (x^2 + 15x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
%t LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 16, 19, 49}, 51] (* _Robert G. Wilson v_, Jul 28 2018 *)
%o (PARI) concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ _Colin Barker_, Jun 08 2018
%o (GAP) a:=[0,1,16,19,49];; 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. A051871, A316672, A317315.
%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), this sequence (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), A303815 (k=29), A316729 (k=30).
%K nonn,easy
%O 0,3
%A _Omar E. Pol_, Jun 06 2018
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