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A035614 Horizontal para-Fibonacci sequence: says which column of Wythoff array (starting column count at 0) contains n+1.

%I A035614 #43 Sep 18 2022 14:31:54
%S A035614 0,1,2,0,3,0,1,4,0,1,2,0,5,0,1,2,0,3,0,1,6,0,1,2,0,3,0,1,4,0,1,2,0,7,
%T A035614 0,1,2,0,3,0,1,4,0,1,2,0,5,0,1,2,0,3,0,1,8,0,1,2,0,3,0,1,4,0,1,2,0,5,
%U A035614 0,1,2,0,3,0,1,6,0,1,2,0,3
%N A035614 Horizontal para-Fibonacci sequence: says which column of Wythoff array (starting column count at 0) contains n+1.
%C A035614 This is probably the same as the "Fibonacci ruler function" mentioned by Knuth. - _N. J. A. Sloane_, Aug 03 2012
%C A035614 From _Amiram Eldar_, Mar 10 2021: (Start)
%C A035614 a(n) is the number of the trailing zeros in the Zeckendorf representation of (n+1) (A014417).
%C A035614 The asymptotic density of the occurrences of k is 1/phi^(k+2), where phi is the golden ratio (A001622).
%C A035614 The asymptotic mean of this sequence is phi. (End)
%D A035614 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 82, solution to Problem 179.
%H A035614 Reinhard Zumkeller, <a href="/A035614/b035614.txt">Table of n, a(n) for n = 0..10000</a>
%H A035614 Casey Mongoven, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from175to192.pdf">Sonification of multiple Fibonacci-related sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
%H A035614 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>
%F A035614 The segment between the first M and the first M+1 is given by the segment before the first M-1.
%F A035614 a(n) = A122840(A014417(n + 1)). - _Indranil Ghosh_, Jun 09 2017
%t A035614 max = 81; wy = Table[(n-k)*Fibonacci[k] + Fibonacci[k+1]*Floor[ GoldenRatio*(n - k + 1)], {n, 1, max}, {k, 1, n}]; a[n_] := Position[wy, n][[1, 2]]-1; Table[a[n], {n, 1, max}] (* _Jean-François Alcover_, Nov 02 2011 *)
%o A035614 (Haskell)
%o A035614 a035614 = a122840 . a014417 . (+ 1)  -- _Reinhard Zumkeller_, Mar 10 2013
%o A035614 (Python)
%o A035614 from sympy import fibonacci
%o A035614 def a122840(n): return len(str(n)) - len(str(int(str(n)[::-1])))
%o A035614 def a014417(n):
%o A035614     k=0
%o A035614     x=0
%o A035614     while n>0:
%o A035614         k=0
%o A035614         while fibonacci(k)<=n: k+=1
%o A035614         x+=10**(k - 3)
%o A035614         n-=fibonacci(k - 1)
%o A035614     return x
%o A035614 def a(n): return a122840(a014417(n + 1)) # _Indranil Ghosh_, Jun 09 2017, after Haskell code by _Reinhard Zumkeller_
%Y A035614 Cf. A000045, A001622, A014417, A019586, A035513, A035612, A122840, A139764.
%K A035614 nonn,nice,easy
%O A035614 0,3
%A A035614 _J. H. Conway_ and _N. J. A. Sloane_