%I A007895
%S A007895 0,1,1,1,2,1,2,2,1,2,2,2,3,1,2,2,2,3,2,3,3,1,2,2,2,3,2,3,3,2,3,3,3,4,1,
%T A007895 2,2,2,3,2,3,3,2,3,3,3,4,2,3,3,3,4,3,4,4,1,2,2,2,3,2,3,3,2,3,3,3,4,2,3,
%U A007895 3,3,4,3,4,4,2,3,3,3,4,3,4,4,3,4,4,4,5,1,2,2,2,3,2,3,3,2,3,3,3,4,2,3,3
%N A007895 Number of terms in Zeckendorf representation of n (write n as a sum of 
               non-consecutive distinct Fibonacci numbers).
%C A007895 Equivalently, number of different Fibonacci numbers that sum up to n: 
               n=sum(Fib(i)) i ranging from 1 to a(n). - Carmine Suriano (surianonoi5(AT)libero.it), 
               Apr 20 2009
%C A007895 a(n) differs from sequence A105446 that counts the symbols in the Roman 
               Fibonacci number representation of n. Example: n=12 Number of symbols 
               in the Roman Fibonacci representing 12 is 2 namely 1A; in this sequnce 
               a(12)=3 since 12=1+3+8=Fib(1)+Fib(4)+Fib(6). Of course a(n)=1 whenever 
               n is a Fibonacci. - Carmine Suriano (surianonoi5(AT)libero.it), Apr 
               20 2009
%C A007895 Also number of 0's (or B's) in the Wythoff representation of n - see 
               the Reble link. See also A135817 for references and links for the 
               Wythoff representation for n>=1. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) 
               Jan 21 2008; N. J. A. Sloane (njas(AT)research.att.com), Jun 28 2008
%C A007895 Or, a(n) = number of applications of Wythoff's B sequence A001950 needed 
               in the unique Wythoff representation of n>=1. E.g. 16=A(B(A(A(B(1))))) 
               = ABAAB = `10110`, hence a(16)=2. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) 
               Jan 21 2008
%C A007895 Let M(0)=0, M(1)=1 and for i > 0, M(i+1)=f(concatenation of M(j), j from 
               0 to i-1) where f is the morphism f(k)=k+1. Then sequence = concatenation 
               of M(j) for j from 0 to infinity. - Claude Lenormand (claude.lenormand(AT)free.fr), 
               Dec 16 2003
%D A007895 D. E. Daykin, Representation of natural numbers as sums of generalized 
               Fibonacci numbers, J. London Math. Soc. 35 (1960) 143-160.
%D A007895 C. G. Lekkerkerker, Voorstelling van natuurlijke getallen door een som 
               van getallen van Fibonacci, Simon Stevin 29 (1952) 190-195.
%D A007895 F. Weinstein, The Fibonacci Partitions, preprint, 1995.
%D A007895 E. Zeckendorf, Representation des nombres naturels par une somme des 
               nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. 
               Liege 41, 179-182, 1972.
%H A007895 T. D. Noe, <a href="b007895.txt">Table of n, a(n) for n=0..10000</a>
%H A007895 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A007895 I. Nemes, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/
               publications/#inemes">Fibonacci representations of multiples of Fibonacci 
               numbers</a>
%H A007895 Don Reble, <a href="a007895.pdf">Zeckendorf vs. Wythoff representations: 
               Comments on A007895</a>
%H A007895 F. V. Weinstein, <a href="http://arXiv.org/abs/math.NT/0307150">Notes 
               on Fibonacci partitions</a>
%F A007895 a(n) = A000120(A003714(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 05 2005
%F A007895 a(n) = A107015(n) + A107016(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 09 2005
%e A007895 a(46) = a(1+3+8+34) = 4.
%Y A007895 Cf. A000045, A035514, A035515, A035516, A035517, A105446.
%Y A007895 Cf. A135817 (lengths of Wythoff representation), A135818 (number of 1's 
               (or A's) in the Wythoff representation).
%Y A007895 Record positions are in A027941.
%Y A007895 Sequence in context: A085761 A102382 A024890 this_sequence A053260 A140223 
               A014643
%Y A007895 Adjacent sequences: A007892 A007893 A007894 this_sequence A007896 A007897 
               A007898
%K A007895 nonn
%O A007895 0,5
%A A007895 Felix Weinstein (wain(AT)ana.unibe.ch) and Clark Kimberling (ck6(AT)evansville.edu)
%E A007895 Edited by N. J. A. Sloane (njas(AT)research.att.com) Jun 27 2008 at the 
               suggestion of R. J. Mathar and Don Reble