### M Theory Lesson 139

Continuing with the wonders of table A033282, the $k = 3$ recursion results in the relation

$F_{n+2}(3) F_{n+1}(1) = \frac{1}{2} F_{n+2}(2) [F_{n+1}(2) + \frac{1}{3} n F_{n+1}(1)]$

For example, considering the codimension 3 edges of the 4d polytope we obtain the relation

$84 \times 9 = \frac{1}{2} \times 56 \times (21 + \frac{2}{3} \times 9)$

Isn't it wonderful how the combinatorics of the associahedra gives us so many relations between integers? One might be forgiven for guessing that operads can tell us something about factorization of an integer into primes.

$F_{n+2}(3) F_{n+1}(1) = \frac{1}{2} F_{n+2}(2) [F_{n+1}(2) + \frac{1}{3} n F_{n+1}(1)]$

For example, considering the codimension 3 edges of the 4d polytope we obtain the relation

$84 \times 9 = \frac{1}{2} \times 56 \times (21 + \frac{2}{3} \times 9)$

Isn't it wonderful how the combinatorics of the associahedra gives us so many relations between integers? One might be forgiven for guessing that operads can tell us something about factorization of an integer into primes.

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