ATRACTOR

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, \[k \geq 2 \rightarrow C_{2}(2k) = 1, 2, 4, 7, 11,...\]