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