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A097840 Chebyshev polynomials S(n,83) + S(n-1,83) with Diophantine property. 4
1, 84, 6971, 578509, 48009276, 3984191399, 330639876841, 27439125586404, 2277116783794691, 188973253929372949, 15682502959354160076, 1301458772372465913359, 108005395603955316648721 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(9*a(n))^2 - 85*b(n)^2 = -4 with b(n)=A097841(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)= S(n, 83) + S(n-1, 83) = S(2*n, sqrt(85)), 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, 83)=A097839(n).

a(n)= (-2/9)*I*((-1)^n)*T(2*n+1, 9*I/2) with the imaginary unit I and Chebyshev's polynomials of the first kind. See the T-triangle A053120.

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

a(n)=83*a(n-1)-a(n-2) for n>1 ; a(0)=1, a(1)=84. [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: A075909 A132052 A273438 * A224177 A223869 A224389

Adjacent sequences:  A097837 A097838 A097839 * A097841 A097842 A097843

KEYWORD

nonn,easy,changed

AUTHOR

Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.