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%I #11 May 25 2015 03:57:56
%S 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,
%T 176,177,178,286,287,288,464,465,466,752,753,754,1218,1219,1220,1972,
%U 1973,1974,3192,3193,3194,5166,5167,5168,8360,8361,8362,13528,13529
%N Numbers k satisfying |k + 1 - 2F| <= 1 for some positive Fibonacci number F.
%C 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.
%H Clark Kimberling, <a href="/A256007/b256007.txt">Table of n, a(n) for n = 0..1000</a>
%F Conjectures from _Colin Barker_, May 24 2015: (Start)
%F a(n) = 2*a(n-3)-a(n-9) for n>12.
%F 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)).
%F (End)
%e F(1) = F(2) contributes {0,1,2}; F(3) contributes {1,2,3}.
%t u = Table[Fibonacci[k], {k, 2, 30}]; Union[2 u - 2, 2 u - 1, 2 u]
%Y Cf. A000045, A001588, A019274.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, May 07 2015
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