A293505 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/2|.

0, 0, 0, 1, 2, 2, 4, 6, 10, 17, 28, 44, 72, 116, 188, 305, 494, 798, 1292, 2090, 3382, 5473, 8856, 14328, 23184, 37512, 60696, 98209, 158906, 257114, 416020, 673134, 1089154, 1762289, 2851444, 4613732, 7465176, 12078908, 19544084, 31622993, 51167078

Offset: 0

Author

Clark Kimberling, Oct 12 2017

Keywords

• nonn
• easy

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 1, -1, -1)

Crossrefs

Cf. A000045, A004695, A293505.

Programs

Mathematica

z = 120; r = 1/2; f[n_] := Fibonacci[n];
Table[Floor[r*f[n]], {n, 0, z}];   (* A004695 *)
Table[Ceiling[r*f[n]], {n, 0, z}]; (* A173173 *)
Table[Round[r*f[n]], {n, 0, z}];   (* A293505 *)

Formula

G.f.: -((x^3 (-1 - x + x^2))/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2))).

a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.

a(n) = floor(1/2 + Fibonacci(n)/2).

a(n) = A004695(n) if (fractional part of Fibonacci(n)/2) < 1/2, otherwise a(n) = A293419(n).