A157737 a(n) = 441*n^2 - 2*n.

439, 1760, 3963, 7048, 11015, 15864, 21595, 28208, 35703, 44080, 53339, 63480, 74503, 86408, 99195, 112864, 127415, 142848, 159163, 176360, 194439, 213400, 233243, 253968, 275575, 298064, 321435, 345688, 370823, 396840, 423739, 451520

Offset

1,1

Comments

The identity (441*n - 1)^2 - (441*n^2 - 2*n)*21^2 = 1 can be written as A158319(n)^2 - a(n)*21^2 = 1 (see Barbeau's paper in link). Also, the identity (388962*n^2 - 1764*n + 1)^2 - (441*n^2 - 2*n)*(18522*n - 42)^2 = 1 can be written as A157739(n)^2 - a(n)*A157738(n)^2 = 1. - Vincenzo Librandi, Jan 25 2012

This last formula is the case s=21 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 05 2012

Links

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

E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (about the first identity in Comments section, row 15 in the first table at p. 85, case d(t) = t*(21^2*t-2)).

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

Formula

G.f.: x*(-439 - 443*x)/(x-1)^3.

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

Mathematica

LinearRecurrence[{3, -3, 1}, {439, 1760, 3963}, 50]

Prog

(Magma) I:=[439, 1760, 3963]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];
(PARI) a(n)=441*n^2-2*n \\ Charles R Greathouse IV, Dec 28 2011

Crossrefs

Cf. A158319, A157738, A157739.

Sequence in context: A142141 A050805 A252135 * A388261 A061328 A177489

Adjacent sequences: A157734 A157735 A157736 * A157738 A157739 A157740

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 05 2009

Status

approved