OFFSET
0,2
COMMENTS
See A265762 for a guide to related sequences.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
FORMULA
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3).
G.f.: (1 + 3 x - 8 x^2 + 2 x^3)/(1 - 2 x - 2 x^2 + x^3).
a(n) = (2^(-1-n)*(-3*(-1)^n*2^(3+n)-(3-sqrt(5))^n*(-7+sqrt(5))+(3+sqrt(5))^n*(7+sqrt(5))))/5 for n>0. - Colin Barker, Sep 29 2016
EXAMPLE
Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[tau,1,1,1,1,...] = sqrt(5) has p(0,x) = -5 + x^2, so a(0) = 1;
[1,tau,1,1,1,...] = (5 + sqrt(5))/5 has p(1,x) = 4 - 10 x + 5 x^2, so a(1) = 5;
[1,1,tau,1,1,...] = (9 - sqrt(5))/4 has p(2,x) = 19 - 18 x + 4 x^2, so a(2) = 4.
MATHEMATICA
PROG
(PARI) Vec((1+3*x-8*x^2+2*x^3)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jan 09 2016
STATUS
approved