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A098252 Chebyshev polynomials S(n,363) + S(n-1,363) with Diophantine property. 3
1, 364, 132131, 47963189, 17410505476, 6319965524599, 2294130074923961, 832762897231873244, 302290637565095063611, 109730668673232276217549, 39831930437745751171906676, 14458881018233034443125905839 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(19*a(n))^2 - 365*b(n)^2 = -4 with b(n)=A098253(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..369

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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

FORMULA

a(n)= S(n, 363) + S(n-1, 363) = S(2*n, sqrt(365)), with S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 363)=A098251(n).

a(n)= (-2/19)*I*((-1)^n)*T(2*n+1, 19*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

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

a(n)=363*a(n-1)-a(n-2), n>1 ; a(0)=1, a(1)=364 . [From 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), ...

MATHEMATICA

LinearRecurrence[{363, -1}, {1, 364}, 20] (* Harvey P. Dale, Feb 03 2015 *)

CROSSREFS

Sequence in context: A140935 A249671 A022196 * A221393 A099113 A073304

Adjacent sequences:  A098249 A098250 A098251 * A098253 A098254 A098255

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified May 26 05:06 EDT 2017. Contains 287074 sequences.