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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). 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 (list; graph; refs; listen; history; text; internal format)
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(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;
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
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
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 999 terms from M. F. Hasler)
S. Mneimneh, Fibonacci in The Curriculum: Not Just a Bad Recurrence, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, Pages 253-258. See Figure 2.
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.
a(n) = abs(A002251(n) - n).
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
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
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
(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
(PARI) A026242=vector(#A002251, n, abs(A002251[n]-n)) \\ M. F. Hasler, Sep 17 2014
CROSSREFS
Cf. A000045 (Fibonacci numbers).
Sequence in context: A085238 A214371 A026338 * A130526 A351955 A174523
KEYWORD
nonn,nice
AUTHOR
STATUS
approved

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Last modified July 18 01:22 EDT 2024. Contains 374377 sequences. (Running on oeis4.)