A157106 a(n) = 5651522*n^2 - 2541672*n + 285769.

3395619, 17808513, 43524451, 80543433, 128865459, 188490529, 259418643, 341649801, 435184003, 540021249, 656161539, 783604873, 922351251, 1072400673, 1233753139, 1406408649, 1590367203, 1785628801, 1992193443, 2210061129, 2439231859, 2679705633, 2931482451

Offset

1,1

Comments

The identity (5651522*n^2 - 2541672*n + 285769)^2 - (1681*n^2 - 756*n + 85)*(137842*n - 30996)^2 = 1 can be written as a(n)^2 - A157010(n)*A157105(n)^2 = 1.

Links

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

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

From Elmo R. Oliveira, Jun 06 2026: (Start)

G.f.: x*(3395619 + 7621656*x + 285769*x^2)/(1 - x)^3.

E.g.f.: -285769 + (285769 + 3109850*x + 5651522*x^2)*exp(x). (End)

Maple

A157106:=n->5651522*n^2 - 2541672*n + 285769; seq(A157106(n), n=1..30); # Wesley Ivan Hurt, Jan 23 2014

Mathematica

LinearRecurrence[{3, -3, 1}, {3395619, 17808513, 43524451}, 30]

Prog

(Magma) I:=[3395619, 17808513, 43524451]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n) = 5651522*n^2 - 2541672*n + 285769

Crossrefs

Cf. A157010, A157105.

Sequence in context: A206168 A206382 A114682 * A123201 A358017 A387886

Adjacent sequences: A157103 A157104 A157105 * A157107 A157108 A157109

Keyword

nonn,easy,changed

Author

Vincenzo Librandi, Feb 23 2009

Status

approved