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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).

%I A026242 #82 Dec 21 2023 15:15:14
%S A026242 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,
%T A026242 19,12,20,21,13,22,14,23,24,15,25,16,26,27,17,28,29,18,30,19,31,32,20,
%U A026242 33,21,34,35,22,36,37,23,38,24,39,40,25,41
%N 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).
%C A026242 Every positive integer occurs exactly twice. a(n) is the parent of n in the tree at A074049. - _Clark Kimberling_, Dec 24 2010
%C A026242 Apparently, if n=F(m) (a Fibonacci number), one of two circumstances arise:
%C A026242 I. a(n)=F(m-1) and a(n-1)=F(m-2). When this happens, a(n) occurs for the first time and a(n-1) occurs for the second time;
%C A026242 II. a(n)=F(m-2) and a(n-1)=F(m-1). When this happens, a(n) occurs for the second time and a(n-1) occurs for the first time. - _Bob Selcoe_, Sep 18 2014
%C A026242 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
%H A026242 Robert G. Wilson v, <a href="/A026242/b026242.txt">Table of n, a(n) for n = 1..10000</a> (first 999 terms from M. F. Hasler)
%H A026242 S. Mneimneh, <a href="https://doi.org/10.1145/2676723.2677215">Fibonacci in The Curriculum: Not Just a Bad Recurrence</a>, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, Pages 253-258. See Figure 2.
%H A026242 Jeffrey Shallit, <a href="/A026242/a026242.pdf">Fibonacci automaton for a(n)</a>
%F A026242 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.
%F A026242 a(n) = abs(A002251(n) - n).
%F A026242 n = a(n) + a(n-1) unless n = A089910(m); if n = A089910(m), then n = a(n) + a(n-1) - m. - _Bob Selcoe_, Sep 20 2014
%F A026242 There is a 17-state automaton that accepts the Zeckendorf (Fibonacci) representation of n and a(n), in parallel.  See the file a026242.pdf. - _Jeffrey Shallit_, Dec 21 2023
%t A026242 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 *)
%o A026242 (PARI) my(A=vector(10^4),i,j=0); while(#A>=i=A000201(j++), A[i]=j; (i=A001950(j))>#A || A[i]=j); A026242=A \\ _M. F. Hasler_, Sep 16 2014 and Sep 18 2014
%o A026242 (PARI) A026242=vector(#A002251,n,abs(A002251[n]-n)) \\ _M. F. Hasler_, Sep 17 2014
%Y A026242 Cf. A026272, A074049, A089910.
%Y A026242 Cf. A000045 (Fibonacci numbers).
%K A026242 nonn,nice
%O A026242 1,3
%A A026242 _Clark Kimberling_