

A026242


a(n) = j if n is L(j), else a(n) = k if n is U(k), where L = A000201, U = A001950 (lower and upper Wythoff sequences).


15



1, 1, 2, 3, 2, 4, 3, 5, 6, 4, 7, 8, 5, 9, 6, 10, 11, 7, 12, 8, 13, 14, 9, 15, 16, 10, 17, 11, 18, 19, 12, 20, 21, 13, 22, 14, 23, 24, 15, 25, 16, 26, 27, 17, 28, 29, 18, 30, 19, 31, 32, 20, 33, 21, 34, 35, 22, 36, 37, 23, 38, 24, 39, 40, 25, 41
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OFFSET

1,3


COMMENTS

Every positive integer occurs exactly twice. a(n) is the parent of n in the tree at A074049.  Clark Kimberling, Dec 24 2010
Apparently, if n=F(m) (a Fibonacci number), one of two circumstances arise:
I. a(n)=F(m1) and a(n1)=F(m2). When this happens, a(n) occurs for the first time and a(n1) occurs for the second time;
II. a(n)=F(m2) and a(n1)=F(m1). When this happens, a(n) occurs for the second time and a(n1) occurs for the first time.  Bob Selcoe, Sep 18 2014
These are the numerators when all fractions, j/r and k/r^2, are arranged in increasing order (where r = golden ratio and j,k are positive integers).  Clark Kimberling, Mar 02 2015


LINKS



FORMULA

a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m; otherwise, a(n) = least positive integer that has not yet occurred.
There is a 17state automaton that accepts the Zeckendorf (Fibonacci) representation of n and a(n), in parallel. See the file a026242.pdf.  Jeffrey Shallit, Dec 21 2023


MATHEMATICA

mx = 100; gr = GoldenRatio; LW[n_] := Floor[n*gr]; UW[n_] := Floor[n*gr^2]; alw = Array[LW, Ceiling[mx/gr]]; auw = Array[UW, Ceiling[mx/gr^2]]; f[n_] := If[ MemberQ[alw, n], Position[alw, n][[1, 1]], Position[auw, n][[1, 1]]]; Array[f, mx] (* Robert G. Wilson v, Sep 17 2014 *)


PROG



CROSSREFS



KEYWORD

nonn,nice


AUTHOR



STATUS

approved



