%I A000119 M0101 N0037
%S A000119 1,1,1,2,1,2,2,1,3,2,2,3,1,3,3,2,4,2,3,3,1,4,3,3,5,2,4,4,2,5,3,3,4,1,4,
%T A000119 4,3,6,3,5,5,2,6,4,4,6,2,5,5,3,6,3,4,4,1,5,4,4,7,3,6,6,3,8,5,5,7,2,6,6,
%U A000119 4,8,4,6,6,2,7,5,5,8,3,6,6,3,7,4,4,5,1,5,5,4,8,4,7,7,3,9,6,6,9,3,8,8,5
%N A000119 Number of representations of n as a sum of distinct Fibonacci numbers.
%C A000119 Number of partitions into distinct Fibonacci parts (1 counted as single 
               Fibonacci number)
%C A000119 Inverse Euler transform of sequence has generating function sum_{n>1} 
               x^F(n)-x^{2F(n)} where F() are the Fibonacci numbers.
%C A000119 A065033(n) = a(A000045(n)).
%D A000119 J. Berstel, An Exercise on Fibonacci Representations, RAIRO/Informatique 
               Theorique, Vol. 35, No 6, 2001, pp. 491-498, in the issue dedicated 
               to Aldo De Luca on the occasion of his 60-th anniversary.
%D A000119 M. Bicknell-Johnson, pp. 53-60 in 'Applications of Fibonacci Numbers', 
               volume 8, ed: F T Howard, Kluwer (1999); see Theorem 3.
%D A000119 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci 
               Association, San Jose, CA, 1972, p. 54.
%D A000119 D. A. Klarner, Representations of N as a sum of distinct elements from 
               special sequences, Fib. Quart., 4 (1966), 289-306 and 322.
%D A000119 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000119 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000119 Paul K. Stockmeyer, "A Smooth Tight Upper Bound for the Fibonacci Representation 
               Function R(N)", Fibonacci Quarterly, Volume 46/47, Number 2, May 
               2009. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 04 2009]
%H A000119 T. D. Noe, <a href="b000119.txt">Table of n, a(n) for n = 0..6765</a>
%H A000119 Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/">Home Page</
               a>
%H A000119 Ron Knott <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
               fibrep.html#sumoffib">Sumthing about Fibonacci Numbers</a>
%H A000119 J. Shallit, <a href="http://www.math.uwaterloo.ca/~shallit/Papers/ntfl.ps">
               Number theory and formal languages</a>, in D. A. Hejhal, J. Friedman, 
               M. C. Gutzwiller and A. M. Odlyzko, eds., Emerging Applications of 
               Number Theory, IMA Volumes in Mathematics and Its Applications, V. 
               109, Springer-Verlag, 1999, pp. 547-570. (Eq. 9.2.)
%F A000119 a(n) = (1/n)*Sum_{k=1..n} b(k)*a(n-k), b(k) = Sum_{f} (-1)^(k/f+1)*f, 
               where the last sum is taken over all Fibonacci numbers f dividing 
               k. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 28 2002
%F A000119 a(n)= 1, if n=0, 1, 2; a(n)= a(fib(i-2)+k)+a(k) if n>2 and 0<=k<=fib(i-3); 
               a(n)= 2*a(k) if n>2 and fib(i-3)<=k<=fib(i-2); a(n)= a(fib(i+1)-2-k) 
               otherwise where fib(i) is largest Fibonacci number (A000045) <= n 
               and k=n-fib(i). [Bicknell-Johnson] - Ron Knott (ron(AT)ronknott.com), 
               Dec 06 2004
%F A000119 a(n) = f(n,1,1) with f(x,y,z) = if x<y then 0^x else f(x-y,y+z,y)+f(x,
               y+z,y). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 11 2009]
%p A000119 with(combinat): p := product((1+x^fibonacci(i)), i=2..25): s := series(p,
               x,1000): for k from 0 to 250 do printf(`%d,`,coeff(s,x,k)) od:
%t A000119 CoefficientList[ Normal@Series[ Product[ 1+z^Fibonacci[ k ], {k, 2, 13} 
               ], {z, 0, 233} ], z ]
%o A000119 (PARI) a(n)=local(A,m,f); if(n<0,0,A=1+x*O(x^n); m=2; while((f=fibonacci(m))<=n,
               A*=1+x^f; m++); polcoeff(A,n))
%Y A000119 Cf. A007000, A003107, A000121. Least inverse is A013583.
%Y A000119 Sequence in context: A160696 A152545 A109967 this_sequence A097368 A109699 
               A029283
%Y A000119 Adjacent sequences: A000116 A000117 A000118 this_sequence A000120 A000121 
               A000122
%K A000119 nonn,nice,new
%O A000119 0,4
%A A000119 N. J. A. Sloane (njas(AT)research.att.com).
%E A000119 More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), 
               May 29 2000