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

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, ___}]]
A112310 = Map[Length, A112309]
(* Peter J. C. Moses, Mar 03 2015 *)

Prog

(Haskell)
a112310 n = a112310_list !! n
a112310_list = concat fss where
   fss = [0] : [1] : (map (map (+ 1))) (zipWith (++) fss $ tail fss)
-- Reinhard Zumkeller, Oct 26 2013

Crossrefs

Number of terms in row n of A112309.

Cf. A003849, A035517, A007895, A007953, A104326, A117479.

Record positions are in A001911. - Ray Chandler, Dec 01 2005

Sequence in context: A128330 A133801 A181630 * A350240 A342739 A137734

Adjacent sequences: A112307 A112308 A112309 * A112311 A112312 A112313

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 01 2005

Extensions

Extended by Ray Chandler, Dec 01 2005

Merged with a sequence from Casey Mongoven, Mar 20 2006, by Franklin T. Adams-Watters, Dec 19 2006

Status

approved