OnyxIonVortex wrote:Wow.
Having obtained the seperate formulae for the number of Alpha and Beta lines in a g-tree of a given k, we calculate its sum:

- a+b.JPG (2.44 KiB) Viewed 1342 times
which works for values greater than 0.
All g-trees treated as graphs with Alpha and Beta lines treated as nodes have an Euler cycle; k=1 has a hamiltionian cycle, k=2 has a hamiltonian path. All graphs of k>2 don't have a hamiltionian path.
The v-name values
As defined before, each member has a distinct v-name. For a given family of n, each v-name of a n+1 family member has 1 more letter, except for n=0. The family of n=1, which consists of m and f, has the
v-number of two, henceforth defined as the sum of the numbers of the letters in the v-names of all the members of a given family or g-tree. Hence, for n=2 v=8, and for n=3 v=24. With this we obtain the formula for the v-number for specific families: it is n*2^n, for n>0.
For whole g-trees of a given k, the formula for the v-number is, as follows:

- v(k).JPG (1.66 KiB) Viewed 1342 times
Palindromic v-names
V-names of certain members are palindromes, i.e. they are the same when read from both the right and normally, from the left. All previously defined border members exhibit this property, but they aren't the only palindromic members. For example, all palindromic members of the family n=3 are 3m, 3m, mfm and fmf, while its non-palindromic ones are 2m-f, 2f-m, m-2f and f-2m. The formula for the number of palindromic members in a family of n is 2^((n+1)/2) for odd n's and 2^(n/2) for n that is even.
The formula for the number of palindromic members of whole g-trees of a given k is given by

- p(k).JPG (6.62 KiB) Viewed 1342 times