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A098262 First differences of Chebyshev polynomials S(n,627)=A098260(n) with Diophantine property. 3
1, 626, 392501, 246097501, 154302740626, 96747572275001, 60660573513685001, 38034082845508220626, 23847309283560140647501, 14952224886709362677762501, 9375021156657486838816440626 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(25*b(n))^2 - 629*a(n)^2 = -4 with b(n)=A098261(n) give all positive solutions of this Pell equation.

LINKS

Table of n, a(n) for n=0..10.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (627, -1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n)= ((-1)^n)*S(2*n, 25*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-627*x+x^2).

a(n)= S(n, 627) - S(n-1, 627) = T(2*n+1, sqrt(629)/2)/(sqrt(629)/2), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.

a(n)=627*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=626 . [From Philippe Deléham, Nov 18 2008]

EXAMPLE

All positive solutions of Pell equation x^2 - 629*y^2 = -4 are

(25=25*1,1), (15700=25*628,626), (9843875=25*393755,392501),

(6172093925=25*246883757,246097501), ...

CROSSREFS

Sequence in context: A045171 A158383 A031728 * A031523 A129974 A031703

Adjacent sequences:  A098259 A098260 A098261 * A098263 A098264 A098265

KEYWORD

nonn,easy,changed

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified March 25 15:14 EDT 2017. Contains 284082 sequences.