%I A137273
%S A137273 1,1,2,3,6,13,37,134,659,4416,41343,546577,10345970,283128770,
%T A137273 11306821624,664047579721,57753201767477,7483309752358051
%N A137273 Number of partitions of nth Fibonacci number into Fibonacci parts obtained 
               by iteratively dividing F(k) into F(n-1) and F(n-2); number of sub-Fibonacci 
               sequences of length n starting with 1,0.
%C A137273 By a sub-Fibonacci sequence we mean a sequence of nonnegative integers 
               b(i) with b(i) <= b(i-1) + b(i-2). Here we are taking b(1) = 1 and 
               b(2) = 0.
%C A137273 In the above, b(i) (for i >= 2) is the number of times F(n-i+2) is divided 
               into the next two smaller Fibonacci numbers in forming the partition.
%e A137273 For the sub-Fibonacci sequence 1,0,1,1,1,2, we split F(6)=8 into 5,3; 
               split the 5 into 3,2; split one 3 into 2,1; and split both 2's into 
               1,1. This gives the partition [3,1^5].
%e A137273 [2^4] is the smallest partition of a Fibonacci number into Fibonacci 
               parts that cannot be obtained in this way.
%o A137273 (PARI) nextfibpart(m) = local(s); s=matsize(m);matrix(s[2],s[1]+s[2]-1,
               i,j,sum(k=max(j-i+1,1),s[1],m[k,i])) alist(n) = {local(v,m); v=vector(n,
               j,1); m=[0;1]; for(i=3,n, m=nextfibpart(m);v[i]=sum(j=1,matsize(m)[1],
               sum(k=1,matsize(m)[2],m[j,k]))); v}
%Y A137273 Cf. A098641, A008934, A002449.
%Y A137273 Sequence in context: A117403 A002877 A065845 this_sequence A135967 A146000 
               A134737
%Y A137273 Adjacent sequences: A137270 A137271 A137272 this_sequence A137274 A137275 
               A137276
%K A137273 nonn
%O A137273 1,3
%A A137273 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 05 2008