A097770 Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n)=A097769(n), n >= 0.

1, 577, 333505, 192765313, 111418017409, 64399421297089, 37222754091700033, 21514687465581321985, 12435452132351912407297, 7187669817811939790095681, 4154460719243168846762896321, 2401271108052733781489163977857

Offset: 0

Author

Wolfdieter Lang, Aug 31 2004

Keywords

• nonn
• easy

Examples

			(x,y) = (12*1=12;1), (6948=12*579;577), (4015932=12*334661;333505), ... give the positive integer solutions to x^2 - 145*y^2 =-1.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..362
• Tanya Khovanova, Recursive Sequences
• Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (578,-1).

Crossrefs

Cf. A097768 for S(n, 486).

Row 12 of array A188647.

Programs

GAP

a:=[1,577];; for n in [3..20] do a[n]:=578*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019

Magma

I:=[1,577]; [n le 2 select I[n] else 578*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019

Mathematica

LinearRecurrence[{578, -1},{1, 577},12] (* Ray Chandler, Aug 12 2015 *)

PARI

my(x='x+O('x^20)); Vec((1-x)/(1-578*x+x^2)) \\ G. C. Greubel, Aug 01 2019

Sage

((1-x)/(1-578*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = S(n, 2*289) - S(n-1, 2*289) = T(2*n+1, sqrt(145))/sqrt(145), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.

a(n) = ((-1)^n)*S(2*n, 24*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.

G.f.: (1-x)/(1-578*x+x^2).

a(n) = 578*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=577. - Philippe Deléham, Nov 18 2008