Friday, June 16, 2017

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 ).

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