A195148 Concentric 20-gonal numbers.

0, 1, 20, 41, 80, 121, 180, 241, 320, 401, 500, 601, 720, 841, 980, 1121, 1280, 1441, 1620, 1801, 2000, 2201, 2420, 2641, 2880, 3121, 3380, 3641, 3920, 4201, 4500, 4801, 5120, 5441, 5780, 6121, 6480, 6841, 7220, 7601, 8000, 8401, 8820, 9241, 9680, 10121

Offset

0,3

Comments

Concentric icosagonal numbers.

Sequence found by reading the line from 0, in the direction 0, 20, ..., and the same line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Main axis, perpendicular to A124080 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) = 5*n^2 + 2*(-1)^n - 2.

a(n) = -a(n-1) + 10*n^2 - 10*n + 1. (End)

G.f.: x*(1+18*x+x^2)/((1+x)*(1-x)^3). - Bruno Berselli, Sep 27 2011

Sum_{n>=1} 1/a(n) = Pi^2/120 + tan(Pi/sqrt(5))*Pi/(8*sqrt(5)). - Amiram Eldar, Jan 17 2023

E.g.f.: 5*x*(x + 1)*cosh(x) + (5*x^2 + 5*x - 4)*sinh(x). - Elmo R. Oliveira, Jun 19 2026

Mathematica

LinearRecurrence[{2, 0, -2, 1}, {0, 1, 20, 41}, 50] (* Harvey P. Dale, Apr 08 2016 *)

Prog

(Magma) [5*n^2+2*(-1)^n-2: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011
(PARI) a(n)=5*n^2+2*(-1)^n-2 \\ Charles R Greathouse IV, Sep 28 2015

Crossrefs

A195322 and A195317 interleaved.

Column 20 of A195040.

Cf. A032527, A032528, A077221, A124080, A195048, A195049, A195142, A195143, A195145, A195146, A195147, A195149, A195162.

Sequence in context: A041790 A041788 A041786 * A171389 A038639 A041792

Adjacent sequences: A195145 A195146 A195147 * A195149 A195150 A195151

Keyword

nonn,easy

Author

Omar E. Pol, Sep 17 2011

Status

approved