OFFSET
1,3
COMMENTS
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.
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.
EXAMPLE
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].
[2^4] is the smallest partition of a Fibonacci number into Fibonacci parts that cannot be obtained in this way.
PROG
(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}
CROSSREFS
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Apr 05 2008
STATUS
approved