A277752 a(n) = Sum_{k=0..n} (-1)^k*floor(phi^k), where phi is the golden ratio (A001622).

1, 0, 2, -2, 4, -7, 10, -19, 27, -49, 73, -126, 195, -326, 516, -848, 1358, -2213, 3564, -5785, 9341, -15135, 24467, -39612, 64069, -103692, 167750, -271454, 439192, -710659, 1149838, -1860511, 3010335, -4870861, 7881181, -12752058, 20633223, -33385298, 54018504, -87403820, 141422306

Offset: 0

Author

Ilya Gutkovskiy, Oct 31 2016

Keywords

• easy
• sign

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).

Crossrefs

Cf. A000032, A000045, A001622, A014217, A020956.

Programs

Mathematica

Accumulate[Table[(-1)^n Floor[GoldenRatio^n], {n, 0, 40}]]
LinearRecurrence[{0, 3, -1, -2, 1}, {1, 0, 2, -2, 4}, 41]

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