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A112310 Number of terms in lazy Fibonacci representation of n. 24
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
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
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.
Record positions are in A001911. - Ray Chandler, Dec 01 2005
Sequence in context: A128330 A133801 A181630 * A350240 A342739 A137734
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

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Last modified March 3 16:27 EST 2024. Contains 370512 sequences. (Running on oeis4.)