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A014641 Odd octagonal numbers: (2n+1)(6n+1). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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, 18 (2015), Article 15.3.2.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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)

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 *)

PROG

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

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

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

CROSSREFS

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

Cf. A160485, A245244.

Sequence in context: A041864 A041866 A020211 * A259677 A089115 A259244

Adjacent sequences:  A014638 A014639 A014640 * A014642 A014643 A014644

KEYWORD

nonn,easy

AUTHOR

Mohammad K. Azarian, Dec 11 1999

EXTENSIONS

More terms from Patrick De Geest

Better description from N. J. A. Sloane

STATUS

approved

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Last modified April 22 06:01 EDT 2021. Contains 343161 sequences. (Running on oeis4.)