A157442 a(n) = 14641*n^2 - 24684*n + 10405.

362, 19601, 68122, 145925, 253010, 389377, 555026, 749957, 974170, 1227665, 1510442, 1822501, 2163842, 2534465, 2934370, 3363557, 3822026, 4309777, 4826810, 5373125, 5948722, 6553601, 7187762, 7851205, 8543930, 9265937

Offset: 1

Author

Vincenzo Librandi, Mar 01 2009

Keywords

• nonn
• easy

Comments

The identity (14641*n^2 - 24684*n + 10405)^2 - (121*n^2 - 204*n + 86)*(1331*n - 1122)^2 = 1 can be written as a(n)^2 - A157440(n)*A157441(n)^2 = 1. - Vincenzo Librandi, Jan 29 2012

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Vincenzo Librandi, X^2-AY^2=1
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A017485, A157440, A157441.

Programs

Magma

I:=[362, 19601, 68122]; [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 29 2012

Mathematica

Table[14641n^2-24684n+10405,{n,30}] (* or *) LinearRecurrence[{3,-3,1},{362,19601,68122},30]

PARI

for(n=1, 40, print1(14641*n^2 - 24684*n + 10405", ")); \\ Vincenzo Librandi, Jan 29 2012

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=362, a(2)=19601, a(3)=68122. - Harvey P. Dale, Oct 22 2011

G.f.: x*(-10405*x^2 - 18515*x - 362)/(x-1)^3. - Harvey P. Dale, Oct 22 2011

a(n) = A017485(11*n-10)^2 + 1. - Bruno Berselli, Jan 29 2012