A157621 a(n) = 625n^2 - 364n + 53.

314, 1825, 4586, 8597, 13858, 20369, 28130, 37141, 47402, 58913, 71674, 85685, 100946, 117457, 135218, 154229, 174490, 196001, 218762, 242773, 268034, 294545, 322306, 351317, 381578, 413089, 445850, 479861, 515122, 551633, 589394, 628405

Offset

1,1

Comments

The identity (781250*n^2-455000*n+66249)^2-(625*n^2-364*n+53)*(31250*n-9100)^2=1 can be written as A157623(n)^2-a(n)*A157622(n)^2=1.

The continued fraction expansion of sqrt(a(n)) is [25n-8; {1, 2, 1, 1, 2, 1, 50n-16}]. - Magus K. Chu, Oct 05 2022

Links

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

Vincenzo Librandi, X^2-AY^2=1 [Dead 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).

G.f.: x*(-314-883*x-53*x^2)/(x-1)^3.

Maple

A157621:= n-> 625*n^2 - 364*n + 53: seq(A157621(n), n=1..40); # Wesley Ivan Hurt, Feb 15 2014

Mathematica

LinearRecurrence[{3, -3, 1}, {314, 1825, 4586}, 40]

Prog

(Magma) I:=[314, 1825, 4586]; [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) = 625*n^2 - 364*n + 53;

Crossrefs

Cf. A157622, A157623.

Sequence in context: A257867 A050818 A050807 * A230945 A183691 A284078

Adjacent sequences: A157618 A157619 A157620 * A157622 A157623 A157624

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 03 2009

Status

approved