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A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property. 5
1, 170, 29069, 4970629, 849948490, 145336221161, 24851643870041, 4249485765555850, 726637214266180309, 124250714153751276989, 21246145483077202184810, 3632966626892047822325521 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(13*b(n))^2 - 173*a(n)^2 = -4 with b(n)=A097845(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..446

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (171, -1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n) = ((-1)^n)*S(2*n, 13*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-171*x+x^2).

a(n) = S(n, 171) - S(n-1, 171) = T(2*n+1, sqrt(173)/2)/(sqrt(173)/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) = 171*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=170 . [Philippe Deléham, Nov 18 2008]

EXAMPLE

All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13=13*1,1), (2236=13*172,170), (382343=13*29411,29069), (65378417=13*5029109,4970629), ...

CROSSREFS

Sequence in context: A210784 A178499 A133328 * A250957 A114048 A187704

Adjacent sequences:  A098241 A098242 A098243 * A098245 A098246 A098247

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified April 28 05:37 EDT 2017. Contains 285557 sequences.