## The dynamics of a trick

### Conjectures II

Conjecture 3

In base $$10$$, the number $$C_{5}(D)$$ of cycles in $$N_{D}$$ with period $$5$$ obeys the recurrence law $C_{5}(2) = 1$ $C_{5}(D+2) = 2C_{5}(D) + 1$ with $$C_{2}(2k+1) = C_{2}(2k)$$ being true for every natural $$k$$. That is, $k \geq 1 \rightarrow C_{5}(2k) = 1, 3, 7, 15, 31, 63,...$

In base $$10$$, the number $$C_{2}(D)$$ of cycles in $$N_{D}$$ with period $$2$$ obeys the recurrence law $C_{2}(4) = 1$ $C_{2}(2(k+1)) = C_{2}(2k) + (k-2).$ That is, $k \geq 2 \rightarrow C_{2}(2k) = 1, 2, 4, 7, 11,...$