Fibonacci Partitions Revisited

Let $F=\{1,2,3,5,8,13,21,34,55,\dots \}$ the sequence of distinct Fibonacci numbers.

The number $1000$ can be decomposed in 15 ways as a sum of elements of $F$

• 5,14
• 3,4,14
• 5,12,13
• 1,2,4,14
• 3,4,12,13
• 5,10,11,13
• 1,2,4,12,13
• 3,4,10,11,13
• 5,8,9,11,13
• 1,2,4,10,11,13
• 3,4,8,9,11,13
• 5,6,7,9,11,13
• 1,2,4,8,9,11,13
• 3,4,6,7,9,11,13
• 1,2,4,6,7,9,11,13

For each decomposition, we give the indexes in $F$ (0-origin):
$\{5,12,13\}\Rightarrow 1000=F_5+F_{12}+F_{13}=13+377+610$

Find the number of decompositions of 1234568 and give the middle one (if there is n decompositions, give the decomposition $\lfloor n/2\rfloor$.
The 1st one has index 0.
The decompositions are sorted by length, then lexicographically.

Answer format: count / (comma delimited list of indexes)

You are given : 15/3,4,10,11,13 for $n=1000$

[My timing: 50 sec]