A099373 Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at 83/2.

2, 83, 6887, 571538, 47430767, 3936182123, 326655685442, 27108485709563, 2249677658208287, 186696137145578258, 15493529705424787127, 1285776269413111753283, 106703936831582850735362, 8855140980751963499281763, 734869997465581387589650967, 60985354648662503206441748498

Offset

0,1

Comments

Used in A099372.

The proper and improper nonnegative solutions of the Pell equation x(n)^2 - 85*y(n)^2 = +4 are x(n) = a(n) and y(n) = 9*A097839(n), n >= 0. - Wolfdieter Lang, Jul 01 2013

Links

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

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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

Formula

a(n) = 83*a(n-1) - a(n-2), n >= 1; a(-1) = 83, a(0) = 2.

a(n) = S(n, 83) - S(n-2, 83) = 2*T(n, 83/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 83) = A097839(n). U-, resp. T-, are Chebyshev polynomials of the second, resp. first, case. See A049310 and A053120.

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

a(n) = ap^n + am^n, with ap := (83+9*sqrt(85))/2 and am := (83-9*sqrt(85))/2.

E.g.f.: 2*exp(83*x/2)*cosh(9*sqrt(85)*x/2). - Stefano Spezia, Apr 06 2023

Example

Pell equation: n=0: 2^2 - 85*0^2 = +4 (improper), n=1: 83^2 - 85*(9*1)^2 = +4, n=2: 6887^2 - 85*(9*83)^2 = +4. - Wolfdieter Lang, Jul 01 2013

Crossrefs

Cf. A049310, A053120, A097839, A099372.

Sequence in context: A225807 A232770 A317724 * A169601 A367900 A205643

Adjacent sequences: A099370 A099371 A099372 * A099374 A099375 A099376

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 18 2004

Status

approved