A157010 a(n) = 1681*n^2 - 756*n + 85.

1010, 5297, 12946, 23957, 38330, 56065, 77162, 101621, 129442, 160625, 195170, 233077, 274346, 318977, 366970, 418325, 473042, 531121, 592562, 657365, 725530, 797057, 871946, 950197, 1031810, 1116785, 1205122, 1296821, 1391882, 1490305

Offset: 1

Author

Vincenzo Librandi, Feb 23 2009

Keywords

• nonn
• easy

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 (A157106(n))^2 - (a(n))*(A157105(n))^2 = 1.

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).

Crossrefs

Cf. A157105, A157106.

Programs

GAP

List([1..40], n-> 1681*n^2 - 756*n + 85); # G. C. Greubel, Feb 23 2019

Magma

I:=[1010, 5297, 12946]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

Maple

A157010:=n->1681*n^2 - 756*n + 85; seq(A157010(n), n=1..30); # Wesley Ivan Hurt, Jan 24 2014

Mathematica

LinearRecurrence[{3,-3,1},{1010,5297,12946},30]
Table[1681n^2-756n+85,{n,40}] (* Harvey P. Dale, Sep 25 2024 *)

PARI

a(n) = 1681*n^2 - 756*n + 85

Sage

[1681*n^2 - 756*n + 85 for n in (1..40)] # G. C. Greubel, Feb 23 2019

Formula

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

G.f: x*(1010 + 2267*x + 85*x^2)/(1-x)^3.

E.g.f.: -85 + (85 + 925*x + 1681*x^2)*exp(x). - G. C. Greubel, Feb 23 2019