A157618 a(n) = 625*n^2 - 886*n + 314.

53, 1042, 3281, 6770, 11509, 17498, 24737, 33226, 42965, 53954, 66193, 79682, 94421, 110410, 127649, 146138, 165877, 186866, 209105, 232594, 257333, 283322, 310561, 339050, 368789, 399778, 432017, 465506, 500245, 536234, 573473, 611962

Offset

1,1

Comments

The identity (781250*n^2 - 1107500*n + 392499)^2 - (625*n^2 - 886*n + 314)*(31250*n - 22150)^2 = 1 can be written as A157620(n)^2 - a(n)*A157619(n)^2 = 1.

The continued fraction expansion of sqrt(a(n)) is [25n-18; {3, 1, 1, 3, 50n-36}]. - Magus K. Chu, Sep 30 2022

Links

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

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

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

Mathematica

Table[625n^2-886n+314, {n, 40}]  (* Harvey P. Dale, Jan 29 2011 *)

Prog

(Magma) I:=[53, 1042, 3281]; [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 - 886*n + 314.

Crossrefs

Cf. A157619, A157620.

Sequence in context: A200532 A289818 A253111 * A093982 A091548 A342899

Adjacent sequences: A157615 A157616 A157617 * A157619 A157620 A157621

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 03 2009

Status

approved