A195317 Centered 40-gonal numbers.

1, 41, 121, 241, 401, 601, 841, 1121, 1441, 1801, 2201, 2641, 3121, 3641, 4201, 4801, 5441, 6121, 6841, 7601, 8401, 9241, 10121, 11041, 12001, 13001, 14041, 15121, 16241, 17401, 18601, 19841, 21121, 22441, 23801, 25201, 26641, 28121, 29641, 31201, 32801, 34441, 36121

Offset: 1

Author

Omar E. Pol, Sep 16 2011

Keywords

• nonn
• easy

Comments

Also centered tetracontagonal numbers or centered tetrakaicontagonal numbers. Also sequence found by reading the line from 1, in the direction 1, 41, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semi-axis opposite to A195322 in the same spiral.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Bisection of A195148.

Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195316, A195318.

Cf. A195162, A195322.

Programs

Magma

[20*n^2 - 20*n + 1: n in [1..50]]; // Vincenzo Librandi, Sep 21 2011

Mathematica

Table[20*n^2 - 20*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

PARI

a(n)=20*n^2-20*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 20*n^2 - 20*n + 1.

Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(5))/(8*sqrt(5)). - Amiram Eldar, Feb 11 2022

G.f.: -x*(1+38*x+x^2)/(x-1)^3. - R. J. Mathar, May 07 2024

From Elmo R. Oliveira, Nov 15 2024: (Start)

E.g.f.: exp(x)*(20*x^2 + 1) - 1.

a(n) = 2*A069133(n) - 1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)