A157759 a(n) = 15780962*n^2 - 25943924*n + 10662963.

500001, 21898963, 74859849, 159382659, 275467393, 423114051, 602322633, 813093139, 1055425569, 1329319923, 1634776201, 1971794403, 2340374529, 2740516579, 3172220553, 3635486451, 4130314273, 4656704019, 5214655689, 5804169283

Offset: 1

Author

Vincenzo Librandi, Mar 06 2009

Keywords

• nonn
• easy

Comments

The identity (15780962*n^2 - 25943924*n + 10662963)^2 - (2809*n^2 - 4618*n+1898)*(297754*n - 244754)^2 = 1 can be written as a(n)^2 - A157757(n)*A157758(n)^2 = 1.

This is the case s=53 and r=2309 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=53, nonnegative r values are: 500, 2309, 3309, 5118, 6118, 7927, 8927, 10736, 11736, ... - 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. A157757, A157758.

Programs

Magma

I:=[500001, 21898963, 74859849]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..30]];

Mathematica

LinearRecurrence[{3, -3, 1}, {500001, 21898963, 74859849}, 30]

PARI

a(n) = 15780962*n^2 - 25943924*n + 10662963;

Formula

G.f.: x*(500001 - 20398960*x - 10662963*x^2)/(1 - x)^3.

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