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

0,3

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).

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

Mathematica

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

Prog

(Magma) [(18*n^2+7*(-1)^n-7)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011
(PARI) a(n)=(18*n^2+7*(-1)^n-7)/4 \\ Charles R Greathouse IV, Sep 28 2015

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

Sequence in context: A041638 A041636 A212428 * A198276 A041640 A041642

Adjacent sequences: A195144 A195145 A195146 * A195148 A195149 A195150

Keyword

nonn,easy

Author

Omar E. Pol, Sep 17 2011

Status

approved