cp's OEIS Frontend

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.

%I A137273 #11 May 27 2025 20:32:19
%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 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 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.
%H A137273 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 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]))
%o A137273 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.
%K A137273 nonn
%O A137273 1,3
%A A137273 _Franklin T. Adams-Watters_, Apr 05 2008