A157514 a(n) = 25*n^2 - n.

24, 98, 222, 396, 620, 894, 1218, 1592, 2016, 2490, 3014, 3588, 4212, 4886, 5610, 6384, 7208, 8082, 9006, 9980, 11004, 12078, 13202, 14376, 15600, 16874, 18198, 19572, 20996, 22470, 23994, 25568, 27192, 28866, 30590, 32364, 34188, 36062

Offset

1,1

Comments

The identity (5000*n^2 - 200*n + 1)^2 - (25*n^2 - n)*(1000*n - 20)^2 = 1 can be written as A157516(n)^2 - a(n)*A157515(n)^2 = 1. This is the case s=5 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Vincenzo Librandi, Jan 26 2012

The continued fraction expansion of sqrt(a(n)) is [5n-1; {1, 8, 1, 10n-2}]. For n=1, this collapses to [4; {1, 8}]. - Magus K. Chu, Sep 21 2022

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).

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012

G.f.: x*(-24 - 26*x)/(x-1)^3. - Vincenzo Librandi, Jan 26 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {24, 98, 222}, 50] (* Vincenzo Librandi, Jan 26 2012 *)

Prog

(Magma) I:=[24, 98, 222]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 26 2012
(PARI) for(n=1, 22, print1(25*n^2 - n", ")); \\ Vincenzo Librandi, Jan 26 2012

Crossrefs

Cf. A157515, A157516.

Sequence in context: A042122 A042124 A042126 * A100152 A233405 A297798

Adjacent sequences: A157511 A157512 A157513 * A157515 A157516 A157517

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 02 2009

Status

approved