%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)).
%C A000119 a(n) = 1 if and only if n+1 is a Fibonacci number. The lengths of such 
               quasi-periods (from Fib(i)-1 to Fib(i+1)-1, inclusive) is a Fibonacci 
               number + 1. The maximum value of a(n) within each subsequent quasi-period 
               increases by a Fibonacci number. For example, from n = 143 to n = 
               232, the maximum is 13. From 232 to 376, the maximum is 16, an increase 
               of 3. From 376 to 609, 21, an increase of 5. From 609 to 986, 26, 
               increasing by 5 again. Each two subsequent maxima seem to increase 
               by the same increment, the next Fibonacci number. [From Kerry Mitchell 
               (lkmitch(AT)gmail.com), Nov 14 2009]
%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