OFFSET
0,1
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.: (9 - 7 x - 35 x^2 + 18 x^3)/(1 - 2 x - 2 x^2 + x^3).
a(n) = (2^(-n)*(-37*(-2)^n-2*(3-sqrt(5))^n*(2+3*sqrt(5))+(3+sqrt(5))^n*(-4+6*sqrt(5))))/5. - Colin Barker, Sep 29 2016
EXAMPLE
Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[1/3,1,1,1,...] = (-1 + 3 sqrt(5))/6 has p(0,x) = -11 + 3 x + 9 x^2, so a(0) = 9;
[1,1/3,1,1,...] = (25 + 9 sqrt(5))/22 has p(1,x) = 5 - 25 x + 11 x^2, so a(1) = 11;
[1,1,1/3,1,...] = (35 - 9 sqrt(5))/10 has p(2,x) = 41 - 35 x + 5 x^2, so a(2) = 5.
MATHEMATICA
PROG
(PARI) a(n) = round((2^(-n)*(-37*(-2)^n-2*(3-sqrt(5))^n*(2+3*sqrt(5))+(3+sqrt(5))^n*(-4+6*sqrt(5))))/5) \\ Colin Barker, Sep 29 2016
(PARI) Vec((9-7*x-35*x^2+18*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
EXTENSIONS
Three typos in data fixed by Colin Barker, Sep 29 2016
STATUS
approved