

A112310


Number of terms in lazy Fibonacci representation of n.


18



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
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OFFSET

0,4


COMMENTS

Equivalently, the number of ones in the maximal Fibonacci bitrepresentation (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 lengthn prefix of the infinite Fibonacci word (A003849). An integer p, 1 <= p <= n, is a period of a lengthn word x if x[i] = x[i+p] for 1 <= i <= np.  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) 18.
Ron Knott, Using the Fibonacci numbers to represent whole numbers.
W. Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 6069.


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. A117479, A035517, A104326, A007895.
Record positions are in A001911.  Ray Chandler, Dec 01 2005
Sequence in context: A128330 A133801 A181630 * A137734 A182210 A078705
Adjacent sequences: A112307 A112308 A112309 * A112311 A112312 A112313


KEYWORD

nonn,easy,changed


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. AdamsWatters, Dec 19 2006


STATUS

approved



