A158321 a(n) = 441n^2 + 2n.

443, 1768, 3975, 7064, 11035, 15888, 21623, 28240, 35739, 44120, 53383, 63528, 74555, 86464, 99255, 112928, 127483, 142920, 159239, 176440, 194523, 213488, 233335, 254064, 275675, 298168, 321543, 345800, 370939, 396960, 423863, 451648

Offset

1,1

Comments

The identity (441*n + 1)^2 - (441*n^2 + 2*n)*21^2 = 1 can be written as A158322(n)^2 - a(n)*21^2 = 1. - Vincenzo Librandi, Jan 24 2012

Also, the identity (388962*n^2 + 1764*n + 1)^2 - (441*n^2 + 2*n)*(18522*n + 42)^2 = 1 can be written as A157741(n)^2 - (n)*A157740(n)^2 = 1 (see the second comment at A157741). - Vincenzo Librandi, Feb 05 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*(21^2*t+2)).

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

Formula

G.f.: x*(443+439*x)/(1-x)^3. - Vincenzo Librandi, Jan 24 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 24 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {443, 1768, 3975}, 50] (* Vincenzo Librandi, Jan 24 2012 *)

Prog

(Magma) I:=[443, 1768, 3975]; [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. \\ Vincenzo Librandi, Jan 24 2012

Crossrefs

Cf. A158322, A157740, A157741.

Sequence in context: A031519 A360827 A031699 * A293978 A345307 A205604

Adjacent sequences: A158318 A158319 A158320 * A158322 A158323 A158324

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 16 2009

Status

approved