A195316 Centered 36-gonal numbers.

1, 37, 109, 217, 361, 541, 757, 1009, 1297, 1621, 1981, 2377, 2809, 3277, 3781, 4321, 4897, 5509, 6157, 6841, 7561, 8317, 9109, 9937, 10801, 11701, 12637, 13609, 14617, 15661, 16741, 17857, 19009, 20197, 21421, 22681, 23977, 25309, 26677, 28081, 29521, 30997, 32509

Offset: 1

Author

Omar E. Pol, Sep 16 2011

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 1, in the direction 1, 37, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195321 in the same spiral.

Links

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

Crossrefs

Bisection of A195147.

Cf. A003154, A069129, A069133, A069190, A195314, A195315, A195317, A195318.

Cf. A069131, A195160, A195321.

Programs

Magma

[(18*n^2-18*n+1): n in [1..50]]; // Vincenzo Librandi, Sep 19 2011

Mathematica

Table[18*n^2 - 18*n + 1, {n, 1, 40}] (* Amiram Eldar, Feb 11 2022 *)

PARI

a(n)=18*n^2-18*n+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 18*n^2 - 18*n + 1.

G.f.: -x*(1 + 34*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 18 2011

Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(7)*Pi/6)/(6*sqrt(7)). - Amiram Eldar, Feb 11 2022

From Elmo R. Oliveira, Nov 14 2024: (Start)

E.g.f.: exp(x)*(18*x^2 + 1) - 1.

a(n) = 2*A069131(n) - 1.

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