OFFSET
0,3
COMMENTS
Alternating sum of A014217.
LINKS
Eric Weisstein's World of Mathematics, Golden Ratio
Index entries for linear recurrences with constant coefficients, signature (0,3,-1,-2,1).
FORMULA
G.f.: (1 - x^2 - x^3)/((1 - x)^2*(1 + 2*x - x^3)).
a(n) = 3*a(n-2) - a(n-3) - 2*a(n-4) + a(n-5).
a(n) = Sum_{k=0..n} (-1)^k*floor(Fibonacci(2k+3)/Fibonacci(k+3)).
a(n) = Sum_{k=0..n} (-1)^k*(L(k) - (1 + (-1)^k)/2), where L(k) is the Lucas numbers beginning at 2 (A000032).
a(n) = 2^(-n-2)*(9*2^n - 2^(n+1)*n - (-2)^n - 2*(1 + sqrt(5))*(sqrt(5) - 1)^n + 2*(sqrt(5) - 1)*(-1-sqrt(5))^n).
a(n) ~ (-1)^n*phi^(n-1).
a(n) = (-1)^n*Lucas(n-1) - (1/4)*(2*n -9 +(-1)^n). - G. C. Greubel, Oct 31 2016
MATHEMATICA
Accumulate[Table[(-1)^n Floor[GoldenRatio^n], {n, 0, 40}]]
LinearRecurrence[{0, 3, -1, -2, 1}, {1, 0, 2, -2, 4}, 41]
CROSSREFS
KEYWORD
easy,sign,changed
AUTHOR
Ilya Gutkovskiy, Oct 31 2016
STATUS
approved