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A097841 First differences of Chebyshev polynomials S(n,83)=A097839(n) with Diophantine property. 5
1, 82, 6805, 564733, 46866034, 3889316089, 322766369353, 26785719340210, 2222891938868077, 184473245206710181, 15309056460218076946, 1270467212952893676337, 105433469618629957059025 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(9*b(n))^2 - 85*a(n)^2 = -4 with b(n)=A097840(n) give all positive solutions of this Pell equation.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..520

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (83, -1).

Index entries for sequences related to Chebyshev polynomials.

FORMULA

a(n)= ((-1)^n)*S(2*n, 9*I) with the imaginary unit I and the S(n, x)=U(n, x/2) Chebyshev polynomials.

G.f.: (1-x)/(1-83*x+x^2).

a(n)= S(n, 83) - S(n-1, 83) = T(2*n+1, sqrt(85)/2)/(sqrt(85)/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)=83*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=82. [From Philippe Deléham, Nov 18 2008]

EXAMPLE

All positive solutions of Pell equation x^2 - 85*y^2 = -4 are

(9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...

MATHEMATICA

CoefficientList[Series[(1 - x)/(1 - 83 x + x^2), {x, 0, 12}], x] (* Michael De Vlieger, Feb 08 2017 *)

CROSSREFS

Sequence in context: A280959 A252705 A239670 * A116123 A116142 A054214

Adjacent sequences:  A097838 A097839 A097840 * A097842 A097843 A097844

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified June 25 18:15 EDT 2017. Contains 288729 sequences.