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

Author

Omar E. Pol, Sep 17 2011

Keywords

• nonn
• easy

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).

Crossrefs

A016802 and A195315 interleaved.

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

Column 16 of A195040.

Programs

Magma

[(8*n^2+3*(-1)^n-3)/2: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

Mathematica

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

PARI

a(n)=(8*n^2+3*(-1)^n-3)/2 \\ Charles R Greathouse IV, Oct 07 2015

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