A195147 Concentric 18-gonal numbers.

0, 1, 18, 37, 72, 109, 162, 217, 288, 361, 450, 541, 648, 757, 882, 1009, 1152, 1297, 1458, 1621, 1800, 1981, 2178, 2377, 2592, 2809, 3042, 3277, 3528, 3781, 4050, 4321, 4608, 4897, 5202, 5509, 5832, 6157, 6498, 6841, 7200, 7561, 7938, 8317, 8712, 9109

Offset: 0

Author

Omar E. Pol, Sep 17 2011

Keywords

• nonn
• easy

Comments

Concentric octadecagonal numbers or concentric octakaidecagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 18, ..., and the same line from 1, in the direction 1, 37, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Main axis, perpendicular to A027468 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

A195321 and A195316 interleaved.

Cf. A032528, A077221, A195142, A195143, A195145, A195146, A195148, A195149.

Cf. A032527, A195047, A195048. Column 18 of A195040. - Omar E. Pol, Sep 29 2011

Programs

Magma

[(18*n^2+7*(-1)^n-7)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {0, 1, 18, 37}, 50] (* Amiram Eldar, Jan 17 2023 *)

PARI

a(n)=(18*n^2+7*(-1)^n-7)/4 \\ Charles R Greathouse IV, Sep 28 2015

Formula

G.f.: -x*(1+16*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011

From Vincenzo Librandi, Sep 27 2011: (Start)

a(n) = (18*n^2 + 7*(-1)^n - 7)/4;

a(n) = -a(n-1) + 9*n^2 - 9*n + 1. (End)

Sum_{n>=1} 1/a(n) = Pi^2/108 + tan(sqrt(7)*Pi/6)*Pi/(6*sqrt(7)). - Amiram Eldar, Jan 17 2023