Bulleye Horizon Cap Count

          Bulleye Horizon Cap Count

Consider this fractal: “Bulleye Horizon”:

How many disks of each size? Let’s call the disks with white on top “white caps”, the others “black caps”. Let W(n) = the number of white caps at stage n, B(n) = the number of black caps, and T(n) = the total number of disks at stage n;  where n = 0 for the whole figure, and each stage is half the size of the previous.

Clearly B(0)=0, W(0)=1, T(0)=1, B(1)=1, W(1)=0, T(1)=1.

Also these recursions apply:

B(n)           =       W(n-1) + 2 W(n-2)

W(n)          =       B(n-1) + 2 B(n-2)

T(n)            =       T(n-1) + 2 T(n-2)

B and W are two entangled semi-Fibonacci sequences. Here is a table of values:

          n             B(n)           W(n)          T(n)

          0                 0                 1                 1

          1                 1                 0                 1

          2                 2                 1                 3

          3                 1                 4                 5

          4                 6                 5                 11

          5                 13               8                 21

          6                 18               25               43

          7                 44               41               85

          8                 91               80               171

          9                 162             179             341

          You can prove that

          T(n)            =                 (2/3)(2^n)  + (1/3)((-1)^n)

          Investigations continue. I have deduced that B and W are linear combinations of powers of 2, -1 and ( (-1/2) +/- i(sqrt(7))/2 ).