A303305 Generalized 17-gonal (or heptadecagonal) numbers: m*(15*m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...

0, 1, 14, 17, 43, 48, 87, 94, 146, 155, 220, 231, 309, 322, 413, 428, 532, 549, 666, 685, 815, 836, 979, 1002, 1158, 1183, 1352, 1379, 1561, 1590, 1785, 1816, 2024, 2057, 2278, 2313, 2547, 2584, 2831, 2870, 3130, 3171, 3444, 3487, 3773, 3818, 4117, 4164, 4476, 4525, 4850

Offset: 0

Author

Omar E. Pol, Jun 06 2018

Keywords

• nonn
• easy

Comments

120*a(n) + 169 is a square. - Bruno Berselli, Jun 08 2018

Partial sums of A317313. - Omar E. Pol, Jul 28 2018

Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 17. - Omar E. Pol, Apr 25 2021

Examples

			From _Omar E. Pol_, Apr 24 2021: (Start)
Illustration of initial terms as vertices of a rectangular spiral:
        43_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _17
         |                                                   |
         |                         0                         |
         |                         |_ _ _ _ _ _ _ _ _ _ _ _ _|
         |                         1                         14
         |
        48
More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5". (End)
		

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).

Crossrefs

Cf. A051869, A194715, A244636, A317313.

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), this sequence (k=17), A274979 (k=18), A303813 (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).

Programs

Mathematica

With[{pp = 17, nn = 55}, {0}~Join~Riffle[Array[PolygonalNumber[pp, #] &, Ceiling[nn/2]], Array[PolygonalNumber[pp, -#] &, Ceiling[nn/2]]]] (* Michael De Vlieger, Jun 06 2018 *)
Table[(30 n (n + 1) + 11 (2 n + 1) (-1)^n - 11)/16, {n, 0, 60}] (* Bruno Berselli, Jun 08 2018 *)
CoefficientList[ Series[-x (x^2 + 13x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 14, 17, 43}, 51] (* Robert G. Wilson v, Jul 28 2018 *)

PARI

concat(0, Vec(x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3) + O(x^40))) \\ Colin Barker, Jun 12 2018

Formula

From Bruno Berselli, Jun 08 2018: (Start)

G.f.: x*(1 + 13*x + x^2)/((1 + x)^2*(1 - x)^3).

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

a(n) = (30*n*(n + 1) + 11*(2*n + 1)*(-1)^n - 11)/16. Therefore:

a(n) = n*(15*n + 26)/8, if n is even, or (n + 1)*(15*n - 11)/8 otherwise.

2*(2*n - 1)*a(n) + 2*(2*n + 1)*a(n-1) - n*(15*n^2 - 13) = 0. (End)