A293553 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/4|.

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 14, 22, 36, 58, 94, 152, 247, 399, 646, 1045, 1691, 2736, 4428, 7164, 11592, 18756, 30348, 49104, 79453, 128557, 208010, 336567, 544577, 881144, 1425722, 2306866, 3732588, 6039454, 9772042, 15811496, 25583539, 41395035, 66978574

Offset: 0

Author

Clark Kimberling, Oct 14 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, A004697, A131132, A293552.

Programs

Mathematica

z = 120; r = 1/4; f[n_] := Fibonacci[n];
Table[Floor[r*f[n]], {n, 0, z}];   (* A004697 *)
Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293552 *)
Table[Round[r*f[n]], {n, 0, z}];   (* A293553 *)

Formula

G.f.: x^4/((-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)/4).

a(n) = A004697(n) if (fractional part of Fibonacci(n)/4) < 1/2, otherwise a(n) = A293552(n).

a(n) = A131132(n-4) for n > 3. - Georg Fischer, Oct 22 2018