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A112310 Number of terms in lazy Fibonacci representation of n.

%I A112310 #42 Aug 28 2025 08:23:42
%S A112310 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 A112310 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 A112310 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 A112310 Number of terms in lazy Fibonacci representation of n.
%C A112310 Equivalently, the number of ones in the maximal Fibonacci bit-representation (A104326) of n.
%C A112310 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 A112310 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 A112310 Reinhard Zumkeller, <a href="/A112310/b112310.txt">Table of n, a(n) for n = 0..10000</a>
%H A112310 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 A112310 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 A112310 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 A112310 a(n) = A007953(A104326(n)). - _Amiram Eldar_, Oct 10 2023
%e A112310 a(10) = 3 because A104326(10) = 1110 contains three ones.
%p A112310 A112310 := proc(n)
%p A112310     convert(A104326(n),base,10) ;
%p A112310     add(d,d=%) ;
%p A112310 end proc:
%p A112310 seq(A112310(n),n=0..120) ; # _R. J. Mathar_, Aug 28 2025
%t A112310 DeleteCases[IntegerDigits[Range[200], 2], {___, 0, 0, ___}]
%t A112310 A112309 = Map[DeleteCases[Reverse[#] Fibonacci[Range[Length[#]] + 1], 0] &, DeleteCases[IntegerDigits[-1 + Range[200], 2], {___, 0, 0, ___}]]
%t A112310 A112310 = Map[Length, A112309]
%t A112310 (* _Peter J. C. Moses_, Mar 03 2015 *)
%o A112310 (Haskell)
%o A112310 a112310 n = a112310_list !! n
%o A112310 a112310_list = concat fss where
%o A112310    fss = [0] : [1] : (map (map (+ 1))) (zipWith (++) fss $ tail fss)
%o A112310 -- _Reinhard Zumkeller_, Oct 26 2013
%Y A112310 Number of terms in row n of A112309.
%Y A112310 Cf. A003849, A035517, A007895, A007953, A104326, A117479.
%Y A112310 Record positions are in A001911. - _Ray Chandler_, Dec 01 2005
%K A112310 nonn,easy,changed
%O A112310 0,4
%A A112310 _N. J. A. Sloane_, Dec 01 2005
%E A112310 Extended by _Ray Chandler_, Dec 01 2005
%E A112310 Merged with a sequence from _Casey Mongoven_, Mar 20 2006, by _Franklin T. Adams-Watters_, Dec 19 2006