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A097770 Pell equation solutions (12*b(n))^2 - 145*a(n)^2 = -1 with b(n)=A097769(n), n>=0. 5
1, 577, 333505, 192765313, 111418017409, 64399421297089, 37222754091700033, 21514687465581321985, 12435452132351912407297, 7187669817811939790095681, 4154460719243168846762896321, 2401271108052733781489163977857 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

EXAMPLE

(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.

MATHEMATICA

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

PROG

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-578*x+x^2)) \\ 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

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

(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

CROSSREFS

Cf. A097768 for S(n, 486).

Row 12 of array A188647.

Sequence in context: A163042 A069365 A163053 * A252377 A252376 A031522

Adjacent sequences:  A097767 A097768 A097769 * A097771 A097772 A097773

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified October 15 17:24 EDT 2019. Contains 328037 sequences. (Running on oeis4.)