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

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, 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, 0, 1, 2, 0, 3, 0, 1, 6, 0, 1, 2, 0, 3

Offset: 0

Author

J. H. Conway and N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

This is probably the same as the "Fibonacci ruler function" mentioned by Knuth. - N. J. A. Sloane, Aug 03 2012

From Amiram Eldar, Mar 10 2021: (Start)

a(n) is the number of the trailing zeros in the Zeckendorf representation of (n+1) (A014417).

The asymptotic density of the occurrences of k is 1/phi^(k+2), where phi is the golden ratio (A001622).

The asymptotic mean of this sequence is phi. (End)

References

• D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 82, solution to Problem 179.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
• N. J. A. Sloane, Classic Sequences

Crossrefs

Cf. A000045, A001622, A014417, A019586, A035513, A035612, A122840, A139764.

Programs

Haskell

a035614 = a122840 . a014417 . (+ 1)  -- Reinhard Zumkeller, Mar 10 2013

Mathematica

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 *)

Python

from sympy import fibonacci
def a122840(n): return len(str(n)) - len(str(int(str(n)[::-1])))
def a014417(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def a(n): return a122840(a014417(n + 1)) # Indranil Ghosh, Jun 09 2017, after Haskell code by Reinhard Zumkeller

Formula

The segment between the first M and the first M+1 is given by the segment before the first M-1.

a(n) = A122840(A014417(n + 1)). - Indranil Ghosh, Jun 09 2017