A137273 Number of partitions of n-th 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.

1, 1, 2, 3, 6, 13, 37, 134, 659, 4416, 41343, 546577, 10345970, 283128770, 11306821624, 664047579721, 57753201767477, 7483309752358051

Offset: 1

Author

Franklin T. Adams-Watters, Apr 05 2008

Keywords

• nonn

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.

Examples

			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.
		

Links

• Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 16.

Crossrefs

Cf. A098641, A008934, A002449.

Programs

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}