A156639 a(n) = 169*n^2 - 140*n + 29.

58, 425, 1130, 2173, 3554, 5273, 7330, 9725, 12458, 15529, 18938, 22685, 26770, 31193, 35954, 41053, 46490, 52265, 58378, 64829, 71618, 78745, 86210, 94013, 102154, 110633, 119450, 128605, 138098, 147929, 158098

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 A156721(n)^2 - a(n)*A156627(n)^2 = 1.

The continued fraction expansion of sqrt(a(n)) is [13n-6; {1, 1, 1, 1, 1, 1, 26n-12}]. - Magus K. Chu, Sep 06 2022

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, A156721.

Programs

Magma

I:=[58, 425, 1130]; [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}, {58, 425, 1130}, 40]
Table[169n^2-140n+29,{n,40}] (* Harvey P. Dale, Mar 24 2023 *)

PARI

a(n)=169*n^2-140*n+29 \\ Charles R Greathouse IV, Dec 23 2011

Formula

G.f.: x*(58 + 251*x + 29*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