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A097837 Chebyshev polynomials S(n,51) + S(n-1,51) with Diophantine property. 4
1, 52, 2651, 135149, 6889948, 351252199, 17906972201, 912904330052, 46540213860451, 2372638002552949, 120957997916339948, 6166485255730784399, 314369790044353664401, 16026692807006306100052, 817046963367277257438251 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(7*a(n))^2 - 53*b(n)^2 = -4 with b(n)=A097838(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..584

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (51, -1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n) = S(n, 51) + S(n-1, 51) = S(2*n, sqrt(53)), 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, 51)=A097836(n).

a(n) = (-2/7)*I*((-1)^n)*T(2*n+1, 7*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

G.f.: (1+x)/(1-51*x+x^2).

a(n) = 51*a(n-1)-a(n-2) ; a(0)=1, a(1)=52. [Philippe Deléham, Nov 18 2008]

EXAMPLE

All positive solutions of Pell equation x^2 - 53*y^2 = -4 are (7=7*1,1), (364=7*52,50), (18557=7*2651,2549), (946043=7*135149,129949), ...

CROSSREFS

Sequence in context: A215595 A134552 A004296 * A247628 A262272 A189908

Adjacent sequences:  A097834 A097835 A097836 * A097838 A097839 A097840

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified August 23 01:14 EDT 2017. Contains 290953 sequences.