A112310 - OEIS

A112310

Number of terms in lazy Fibonacci representation of n.

0, 1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 4, 5, 3, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 6, 4, 4, 5, 4, 5, 5, 5, 6, 4, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 7, 4, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 7, 5, 5, 6, 5, 6, 6, 6, 7, 5, 6, 6, 6, 7, 6, 6, 7, 6, 7, 7, 7, 8, 5, 5, 6, 5, 6, 6, 6, 7, 5, 6, 6, 6, 7, 6, 6, 7, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

Equivalently, the number of ones in the maximal Fibonacci bit-representation (A104326) of n.

Conjecture: if we split the sequence in groups that contain Fibonacci(k) terms like (0), (1), (1, 2), (2, 2, 3), (2, 3, 3, 3, 4), (3, 3, 4, 3, 4, 4, 4, 5) etc, the sums in the groups are the terms of A023610. - Gary W. Adamson, Nov 02 2010

Equivalently, the number of periods in the length-n prefix of the infinite Fibonacci word (A003849). An integer p, 1 <= p <= n, is a period of a length-n word x if x[i] = x[i+p] for 1 <= i <= n-p. - Jeffrey Shallit, May 23 2020

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly 3, No. 1 (1965), 1-8.

Ron Knott, Using the Fibonacci numbers to represent whole numbers.

Wolfgang Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 60-69.

FORMULA

a(n) = A007953(A104326(n)). - Amiram Eldar, Oct 10 2023

EXAMPLE

a(10) = 3 because A104326(10) = 1110 contains three ones.

MATHEMATICA

DeleteCases[IntegerDigits[Range[200], 2], {___, 0, 0, ___}]

A112309 = Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]] + 1], 0] &, DeleteCases[IntegerDigits[-1 + Range[200], 2], {___, 0, 0, ___}]]