A158493 a(n) = 20*n^2 + 1.

1, 21, 81, 181, 321, 501, 721, 981, 1281, 1621, 2001, 2421, 2881, 3381, 3921, 4501, 5121, 5781, 6481, 7221, 8001, 8821, 9681, 10581, 11521, 12501, 13521, 14581, 15681, 16821, 18001, 19221, 20481, 21781, 23121, 24501, 25921, 27381, 28881, 30421, 32001, 33621, 35281

Offset

0,2

Comments

The identity (20*n^2 + 1)^2 - (100*n^2 + 10)*(2*n)^2 = 1 can be written as a(n)^2 - A158492(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 21 2012

Sequence found by reading the segment (1, 21) together with the line from 21, in the direction 21, 81, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - Omar E. Pol, Nov 05 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]

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

Formula

From Vincenzo Librandi, Feb 21 2012: (Start)

G.f.: -(1 + 18*x + 21*x^2)/(x-1)^3.

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

From Amiram Eldar, Mar 06 2023: (Start)

Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/2.

Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/2. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 21, 81}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

Prog

(Magma) I:=[1, 21, 81]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 21 2012
(PARI) for(n=0, 40, print1(20*n^2 + 1", ")); \\ Vincenzo Librandi, Feb 21 2012

Crossrefs

Cf. A005843, A158492, A195162.

Sequence in context: A296970 A068085 A135945 * A159743 A219593 A195961

Adjacent sequences: A158490 A158491 A158492 * A158494 A158495 A158496

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 21 2009

Extensions

Edited by N. J. A. Sloane, Oct 12 2009

Status

approved