A195146 Concentric 16-gonal numbers.

0, 1, 16, 33, 64, 97, 144, 193, 256, 321, 400, 481, 576, 673, 784, 897, 1024, 1153, 1296, 1441, 1600, 1761, 1936, 2113, 2304, 2497, 2704, 2913, 3136, 3361, 3600, 3841, 4096, 4353, 4624, 4897, 5184, 5473, 5776, 6081, 6400, 6721, 7056, 7393, 7744, 8097, 8464

Offset

0,3

Comments

Concentric hexadecagonal numbers or concentric hexakaidecagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 16, ..., and the same line from 1, in the direction 1, 33, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Main axis, perpendicular to A033996 in the same spiral.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

From Vincenzo Librandi, Sep 27 2011: (Start)

a(n) = (8*n^2 + 3*(-1)^n - 3)/2;

a(n) = -a(n-1) + 8*n^2 - 8*n + 1. (End)

G.f. -x*(1+14*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011

Sum_{n>=1} 1/a(n) = Pi^2/96 + tan(sqrt(3)*Pi/4)*Pi/(8*sqrt(3)). - Amiram Eldar, Jan 16 2023

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {0, 1, 16, 33}, 50] (* Amiram Eldar, Jan 16 2023 *)

Prog

(Magma) [(8*n^2+3*(-1)^n-3)/2: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011
(PARI) a(n)=(8*n^2+3*(-1)^n-3)/2 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

A016802 and A195315 interleaved.

Cf. A032528, A033996, A074377, A077221, A195142, A195143, A195145, A195147, A195148, A195149.

Column 16 of A195040.

Sequence in context: A041500 A361692 A132758 * A365802 A198275 A041504

Adjacent sequences: A195143 A195144 A195145 * A195147 A195148 A195149

Keyword

nonn,easy

Author

Omar E. Pol, Sep 17 2011

Status

approved