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A097735 Pell equation solutions (8*a(n))^2 - 65*b(n)^2 = -1 with b(n):=A097736(n), n>=0. 2
1, 259, 66821, 17239559, 4447739401, 1147499525899, 296050429942541, 76379863425649679, 19705708713387674641, 5083996468190594407699, 1311651383084459969511701, 338400972839322481539611159 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..11.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (258, -1).

FORMULA

a(n)= S(n, 2*129) + S(n-1, 2*129) = S(2*n, 2*sqrt(65)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).

a(n)= ((-1)^n)*T(2*n+1, 8*I)/(8*I) with the imaginary unit I and Chebyshev polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-2*129*x+x^2).

a(n)=258*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=259 . [From Philippe Deléham, Nov 18 2008]

EXAMPLE

(x,y) = (8,1), (2072,257), (534568,66305), ... give the positive integer solutions to x^2 - 65*y^2 =-1.

MATHEMATICA

LinearRecurrence[{258, -1}, {1, 259}, 20] (* Harvey P. Dale, Oct 30 2011 *)

CROSSREFS

Cf. A097731 for S(n, 2*129).

Sequence in context: A229433 A022221 A121918 * A063485 A252248 A214471

Adjacent sequences:  A097732 A097733 A097734 * A097736 A097737 A097738

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 31 2004

STATUS

approved

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Last modified December 5 16:07 EST 2016. Contains 278770 sequences.