A097841 First differences of Chebyshev polynomials S(n,83) = A097839(n) with Diophantine property.

1, 82, 6805, 564733, 46866034, 3889316089, 322766369353, 26785719340210, 2222891938868077, 184473245206710181, 15309056460218076946, 1270467212952893676337, 105433469618629957059025

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

Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.

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. - 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)/(1-83x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{83, -1}, {1, 82}, 20] (* G. C. Greubel, Jan 13 2019 *)

Prog

(PARI) my(x='x+O('x^20)); Vec((1-x)/(1-83*x+x^2)) \\ G. C. Greubel, Jan 13 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019
(Sage) ((1-x)/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019
(GAP) a:=[1, 82];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019

Crossrefs

Sequence in context: A252705 A239670 A292423 * A116123 A116142 A054214

Adjacent sequences: A097838 A097839 A097840 * A097842 A097843 A097844

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 10 2004

Status

approved