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

0,3

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.

Links

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

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)

Example

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

Mathematica

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

Crossrefs

Cf. A000045, A001588, A019274.

Sequence in context: A022773 A049997 A226857 * A324689 A280578 A303878

Adjacent sequences: A256004 A256005 A256006 * A256008 A256009 A256010

Keyword

nonn,easy

Author

Clark Kimberling, May 07 2015

Status

approved