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The Collatz variations pattern seems not to have any recurrence relationbetween numbers. But knowing that there is at least a natural number thatconverges after several iterations we construct a function $f_{X,Y}$ that isequal to the value of convergence for all convergent sequences. A canonicaldecomposition can be expressed for such numbers.

Canonical decomposition in $\mathbb{r}_+^*$ of a convergent natural number by the collatz iterations