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We give a new construction of the Kimberling sequence defined by:(a)1 belongs to S;(b)if the positive integer x belongs to S, then 2x and 4x-1 belong to S; and(c)nothing else belongs to S,hence, S=12346781112141516…which is sequence A052499 in the Sloane's On-line Encyclopedia of Integer Sequences, by proving that this sequence is equal to sequence 1+ A003754, the sequence of integers whose binary...
Define a Wythoff's sequence as a sequence of pairs of integers {(An,Bn)}n>n0 such that there exists a finite set of integers T, An=mex({Ai,Bi:i<n}∪T), Bn-An=n, and {Bn}∩T=∅. Structural properties and behaviors of Wythoff's sequence are investigated. The main result is that for such a sequence, there always exists an integer α such that when n is large enough, |An-⌊nφ⌋-α|⩽1, where φ=(1+5)/2,...
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