A256007 - OEIS

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

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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