A137273 - OEIS

%I #8 Dec 10 2024 12:32:26

%S 1,1,2,3,6,13,37,134,659,4416,41343,546577,10345970,283128770,

%T 11306821624,664047579721,57753201767477,7483309752358051

%N 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.

%C 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 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.

%H Olivier Danvy, <a href="https://arxiv.org/abs/2412.03127">Summa Summarum: Moessner's Theorem without Dynamic Programming</a>, arXiv:2412.03127 [cs.DM], 2024. See p. 16.

%e 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 [2^4] is the smallest partition of a Fibonacci number into Fibonacci parts that cannot be obtained in this way.

%o (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 Cf. A098641, A008934, A002449.

%K nonn

%O 1,3

%A _Franklin T. Adams-Watters_, Apr 05 2008