A156721 a(n) = 57122*n^2 - 47320*n + 9801.

19603, 143649, 381939, 734473, 1201251, 1782273, 2477539, 3287049, 4210803, 5248801, 6401043, 7667529, 9048259, 10543233, 12152451, 13875913, 15713619, 17665569, 19731763, 21912201, 24206883, 26615809, 29138979

Offset: 1

Author

Vincenzo Librandi, Feb 15 2009

Keywords

• nonn
• easy

Comments

The identity (57122*n^2 - 47320*n+9801)^2 - (169*n^2 - 140*n + 29)*(4394*n - 1820)^2 = 1 can be written as a(n)^2 - A156639(n)*A156627(n)^2 = 1.

This is the case s=13 and r=70 of the identity (2*(s^2*n-r)^2+1)^2 - (((s^2*n-r)^2+1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2+1)/s^2 is an integer if r^2 == -1 (mod s^2). Therefore, for s=13, nonnegative r values are: 70, 99, 239, 268, 408, 437, 577, 606, 746, 775, 915, 944, ... - Bruno Berselli, Apr 24 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A156627, A156639.

Programs

Magma

I:=[19603, 143649, 381939]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]];

Mathematica

LinearRecurrence[{3, -3, 1}, {19603, 143649, 381939}, 40]

PARI

a(n)=57122*n^2-47320*n+9801 \\ Charles R Greathouse IV, Dec 23 2011

Formula

G.f.: x*(19603 + 84840*x + 9801*x^2)/(1 - x)^3.

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

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010