A181679 a(n) = 121*n^2 + 2*n.

123, 488, 1095, 1944, 3035, 4368, 5943, 7760, 9819, 12120, 14663, 17448, 20475, 23744, 27255, 31008, 35003, 39240, 43719, 48440, 53403, 58608, 64055, 69744, 75675, 81848, 88263, 94920, 101819, 108960, 116343, 123968, 131835, 139944, 148295, 156888, 165723, 174800, 184119, 193680

Offset

1,1

Comments

The identity (121*n + 1)^2 - (121*n^2 + 2*n)*11^2 = 1 can be written as A158131(n)^2 - a(n)*11^2 = 1 (see Barbeau's paper in link).

Also, the identity (29282*n^2 + 484*n + 1)^2 - (121*n^2 + 2*n)*(2662*n + 22)^2 = 1 can be written as A157614(n)^2 - a(n)*A157613(n)^2 = 1. - Vincenzo Librandi, Feb 21 2012

This last formula is the case s=11 of the identity (2*s^4*n^2 + 4*s^2*n + 1)^2 - (s^2*n^2 + 2*n)*(2*s^3*n + 2*s)^2 = 1. - Bruno Berselli, Feb 21 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Vincenzo Librandi, X^2-AY^2=1

E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(11^2*t+2)).

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

Formula

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

G.f.: x*(-119*x - 123)/(x-1)^3.

Mathematica

LinearRecurrence[{3, -3, 1}, {123, 488, 1095}, 50]

Prog

(Magma) [ n*(121*n+2): n in [1..40] ];
(PARI) a(n) = n*(121*n+2).

Crossrefs

Cf. A157613, A157614, A158131.

Sequence in context: A091331 A248556 A031689 * A193251 A074303 A077379

Adjacent sequences: A181676 A181677 A181678 * A181680 A181681 A181682

Keyword

nonn,easy

Author

Vincenzo Librandi, Nov 18 2010, Nov 19 2010

Status

approved