A256007 Numbers k satisfying |k + 1 - 2F| <= 1 for some positive Fibonacci number F.

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 16, 24, 25, 26, 40, 41, 42, 66, 67, 68, 108, 109, 110, 176, 177, 178, 286, 287, 288, 464, 465, 466, 752, 753, 754, 1218, 1219, 1220, 1972, 1973, 1974, 3192, 3193, 3194, 5166, 5167, 5168, 8360, 8361, 8362, 13528, 13529

Offset: 0

Author

Clark Kimberling, May 07 2015

Keywords

• nonn
• easy

Comments

For r > 0, define f(n) = floor(n*r) if n is odd and f(n) = floor(n/r) if n is even. Let S(r,n) be the set {n, f(n), f(f(n)), ...} of iterates of f starting with n. Conjecture: if r = (1 + sqrt(5))/2, then S(r,n) is bounded if and only if n is in this sequence.

Examples

			F(1) = F(2) contributes {0,1,2}; F(3) contributes {1,2,3}.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000045, A001588, A019274.

Programs

Mathematica

u = Table[Fibonacci[k], {k, 2, 30}]; Union[2 u - 2, 2 u - 1, 2 u]

Formula

Conjectures from Colin Barker, May 24 2015: (Start)

a(n) = 2*a(n-3)-a(n-9) for n>12.

G.f.: -x*(x^11+x^10+x^9+2*x^8+x^7-x^4-2*x^3-3*x^2-2*x-1) / ((x-1)*(x^2+x+1)*(x^6+x^3-1)).

(End)