

A099373


Twice Chebyshev's polynomials of the first kind, T(n,x), evaluated at 83/2.


3



2, 83, 6887, 571538, 47430767, 3936182123, 326655685442, 27108485709563, 2249677658208287, 186696137145578258, 15493529705424787127, 1285776269413111753283, 106703936831582850735362
(list;
graph;
refs;
listen;
history;
text;
internal format)



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(n1)  a(n2), n >= 1; a(1) = 83, a(0) = 2.
a(n) = S(n, 83)  S(n2, 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's polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (283*x)/(183*x+x^2).
a(n) = ap^n + am^n, with ap := (83+9*sqrt(85))/2 and am := (839*sqrt(85))/2.


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

Sequence in context: A225807 A232770 A317724 * A169601 A205643 A215263
Adjacent sequences: A099370 A099371 A099372 * A099374 A099375 A099376


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Oct 18 2004


STATUS

approved



