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 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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Oct 18 2004
STATUS
approved