OFFSET
0,2
COMMENTS
Partial sums of A169986.
LINKS
Eric Weisstein's World of Mathematics, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).
FORMULA
G.f.: (1 + x - x^2 - x^3 - x^4)/((1 - x)^2*(1 - 2*x^2 - x^3)).
a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5).
a(n) = (10*n - 5*(-1)^n + 2^(1-n)*sqrt(5)*(5 + 3*sqrt(5))*(1 + sqrt(5))^n + sqrt(5)*2^(1-n)*(3*sqrt(5) - 5) *(1 - sqrt(5))^n - 35)/20.
a(n) ~ phi^(n+2).
MATHEMATICA
Accumulate[Table[Ceiling[GoldenRatio^n], {n, 0, 40}]]
LinearRecurrence[{2, 1, -3, 0, 1}, {1, 3, 6, 11, 18}, 41]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Dec 06 2016
STATUS
approved