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A098253 First differences of Chebyshev polynomials S(n,363)=A098251(n) with Diophantine property. 4
1, 362, 131405, 47699653, 17314842634, 6285240176489, 2281524869222873, 828187242287726410, 300629687425575463957, 109127748348241605689981, 39613072020724277289999146, 14379436015774564414664000017 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(19*b(n))^2 - 365*a(n)^2 = -4 with b(n)=A098252(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..389

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (363,-1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: (1 - x)/(1 - 363*x + x^2).

a(n) = ((-1)^n)*S(2*n, 19*i) with the imaginary unit i and the S(n, x)=U(n, x/2) Chebyshev polynomials.

a(n) = S(n, 363) - S(n-1, 363) = T(2*n+1, sqrt(365)/2)/(sqrt(365)/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) = 363*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=362. - Philippe Deléham, Nov 18 2008

EXAMPLE

All positive solutions of Pell equation x^2 - 365*y^2 = -4 are

(19=19*1,1), (6916=19*364,362), (2510489=19*132131,131405),

(911300591=19*47963189,47699653), ...

CROSSREFS

Sequence in context: A200558 A212324 A007565 * A031517 A192449 A116285

Adjacent sequences:  A098250 A098251 A098252 * A098254 A098255 A098256

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified April 27 00:46 EDT 2017. Contains 285506 sequences.