A157105 a(n) = 137842*n - 30996.

106846, 244688, 382530, 520372, 658214, 796056, 933898, 1071740, 1209582, 1347424, 1485266, 1623108, 1760950, 1898792, 2036634, 2174476, 2312318, 2450160, 2588002, 2725844, 2863686, 3001528, 3139370, 3277212, 3415054

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 - (A157010(n))*(a(n))^2 = 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A157010, A157106.

Programs

Magma

I:=[106846, 244688, 382530]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

Mathematica

LinearRecurrence[{2,-1},{106846,244688},30] (* Harvey P. Dale, Mar 31 2013 *)
82*(1681*Range[30] -378) (* G. C. Greubel, Jan 11 2022 *)

PARI

a(n) = 137842*n - 30996

Sage

[82*(1681*n - 378) for n in (1..30)] # G. C. Greubel, Jan 11 2022

Formula

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

G.f: 82*x*(1303 + 378*x)/(1-x)^2.

E.g.f.: 82*(378 - (378 - 1681*x)*exp(x)). - G. C. Greubel, Jan 11 2022