A014641 Odd octagonal numbers: (2n+1)*(6n+1).

1, 21, 65, 133, 225, 341, 481, 645, 833, 1045, 1281, 1541, 1825, 2133, 2465, 2821, 3201, 3605, 4033, 4485, 4961, 5461, 5985, 6533, 7105, 7701, 8321, 8965, 9633, 10325, 11041, 11781, 12545, 13333, 14145, 14981, 15841, 16725, 17633, 18565, 19521, 20501, 21505

Offset: 0

Author

Mohammad K. Azarian, Dec 11 1999

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5000
• Richard P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014.
• Richard P. Brent, Generalising Tuenter's binomial sums, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.2.
• Leo Tavares, Illustration: Square Block Stars
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000567, A001082, A014642, A014793, A014794, A243201, A289873.

Cf. A160485, A245244.

Cf. A016754, A014105.

Programs

GAP

List([0..50],n->(2*n+1)*(6*n+1)); # Muniru A Asiru, Feb 05 2019

Magma

[ (2*n+1)*(6*n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 08 2014

Maple

A014641:=n->(2*n+1)*(6*n+1); seq(A014641(n), n=0..50); # Wesley Ivan Hurt, Jun 08 2014

Mathematica

Table[(2n + 1)(6n + 1), {n, 0, 49}]  (* Harvey P. Dale, Mar 24 2011 *)

PARI

a(n)=(2*n+1)*(6*n+1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = a(n-1) + 24*n - 4, with n > 0, a(0)=1. - Vincenzo Librandi, Dec 28 2010

G.f.: (1 + 18*x + 5*x^2)/(1 - 3*x + 3*x^2 - x^3). - Colin Barker, Jan 06 2012

a(n) = A289873(6*n+2). - Hugo Pfoertner, Jul 15 2017

From Peter Bala, Jan 22 2018: (Start)

This is the polynomial Qbar(2,n) in Brent. See A160485 for the triangle of coefficients (with signs) of the Qbar polynomials.

a(n) = (1/4^n) * Sum_{k = 0..n} (2*k + 1)^4*binomial(2*n + 1, n - k).

a(n-1) = (2/4^n) * binomial(2*n,n) * ( 1 + 3^4*(n - 1)/(n + 1) + 5^4*(n - 1)*(n - 2)/((n + 1)*(n + 2)) + 7^4*(n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)) + ... ). (End)

From Amiram Eldar, Feb 27 2022: (Start)

Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi + 3*log(3))/8.

Sum_{n>=0} (-1)^n/a(n) = Pi/8 + sqrt(3)*log(2+sqrt(3))/4. (End)

E.g.f.: exp(x)*(1 + 20*x + 12*x^2). - Stefano Spezia, Apr 16 2022

a(n) = A016754(n) + 4*A014105(n). - Leo Tavares, May 20 2022

Extensions

More terms from Patrick De Geest

Better description from N. J. A. Sloane