A112310 - OEIS

%I #40 Jan 05 2025 19:51:38

%S 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,

%T 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,

%U 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

%N Number of terms in lazy Fibonacci representation of n.

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

%C 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

%C 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

%H Reinhard Zumkeller, <a href="/A112310/b112310.txt">Table of n, a(n) for n = 0..10000</a>

%H J. L. Brown, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-1/brown.pdf">A new characterization of the Fibonacci numbers</a>, Fibonacci Quarterly 3, No. 1 (1965), 1-8.

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrep.html">Using the Fibonacci numbers to represent whole numbers</a>.

%H Wolfgang Steiner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-1/paper43-1-8.pdf">The joint distribution of greedy and lazy Fibonacci expansions</a>, Fib. Q., 43 (No. 1, 2005), 60-69.

%F a(n) = A007953(A104326(n)). - _Amiram Eldar_, Oct 10 2023

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

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

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

%t A112310 = Map[Length, A112309]

%t (* _Peter J. C. Moses_, Mar 03 2015 *)

%o (Haskell)

%o a112310 n = a112310_list !! n

%o a112310_list = concat fss where

%o    fss = [0] : [1] : (map (map (+ 1))) (zipWith (++) fss $ tail fss)

%o -- _Reinhard Zumkeller_, Oct 26 2013

%Y Number of terms in row n of A112309.

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

%Y Record positions are in A001911. - _Ray Chandler_, Dec 01 2005

%K nonn,easy,changed

%O 0,4

%A _N. J. A. Sloane_, Dec 01 2005

%E Extended by _Ray Chandler_, Dec 01 2005

%E Merged with a sequence from _Casey Mongoven_, Mar 20 2006, by _Franklin T. Adams-Watters_, Dec 19 2006