OFFSET
0,1
COMMENTS
a(n) and b(n):= A097836(n-1) with b(0) = 0 are the improper and proper nonnegative solutions of the Pell equation a(n)^2 - 53*(7*b(n))^2 = +4. - Wolfdieter Lang, Jun 27 2013
LINKS
Indranil Ghosh, Table of n, a(n) for n = 0..584
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (51, -1).
FORMULA
a(n) = 51*a(n-1) - a(n-2), n >= 1; a(-1)=51, a(0)=2.
a(n) = S(n, 51) - S(n-2, 51) = 2*T(n, 51/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 51)=A097836(n). U-, resp. T-, are Chebyshev polynomials of the second, resp. first, case. See A049310 and A053120.
a(n)= ap^n + am^n, with ap:=(51 + 7*sqrt(53))/2 and am:=(51 - 7*sqrt(53))/2.
G.f.: (2-51*x)/(1-51*x+x^2).
MATHEMATICA
LinearRecurrence[{51, -1}, {2, 51}, 15] (* or *) CoefficientList[Series[(2 - 51 x)/(1 - 51 x + x^2), {x, 0, 14}], x] (* Michael De Vlieger, Feb 08 2017 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Oct 18 2004
STATUS
approved