A303813 Generalized 19-gonal (or enneadecagonal) numbers: m*(17*m - 15)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 16, 19, 49, 54, 99, 106, 166, 175, 250, 261, 351, 364, 469, 484, 604, 621, 756, 775, 925, 946, 1111, 1134, 1314, 1339, 1534, 1561, 1771, 1800, 2025, 2056, 2296, 2329, 2584, 2619, 2889, 2926, 3211, 3250, 3550, 3591, 3906, 3949, 4279, 4324, 4669, 4716, 5076, 5125, 5500, 5551, 5941, 5994, 6399

Offset

0,3

Comments

Numbers k for which 136*k + 225 is a square. - Bruno Berselli, Jul 10 2018

Partial sums of A317315. - Omar E. Pol, Jul 28 2018

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

From Colin Barker, Jun 08 2018: (Start)

G.f.: x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2).

a(n) = (34*n^2 + 60*n)/16 for n even.

a(n) = (34*n^2 + 8*n - 26)/16 for n odd.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

(End)

Mathematica

With[{nn = 54}, {0}~Join~Riffle[Array[PolygonalNumber[19, #] &, Ceiling[nn/2]], Array[PolygonalNumber[19, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
CoefficientList[ Series[-x (x^2 + 15x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 16, 19, 49}, 51] (* Robert G. Wilson v, Jul 28 2018 *)

Prog

(PARI) concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jun 08 2018
(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

Crossrefs

Cf. A051871, A316672, A317315.

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).

Sequence in context: A034012 A086121 A030402 * A335210 A061934 A195683

Adjacent sequences: A303810 A303811 A303812 * A303814 A303815 A303816

Keyword

nonn,easy

Author

Omar E. Pol, Jun 06 2018

Status

approved