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

0,3

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A363930 A361755 A362755 * A212138 A133735 A238801

Adjacent sequences: A035611 A035612 A035613 * A035615 A035616 A035617

Keyword

nonn,nice,easy

Author

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

Status

approved