Gold Horizon Cap Count

          Gold Horizon Cap Count

Consider this fractal: “Gold 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 1/phi times 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) + W(n-2)

W(n)          =       B(n-1) + B(n-2)

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

T is the Fibonnacci sequence, and B and W are two entangled Fibonacci sequences. Here is a table of values:

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

          0                 0                 1                 1

          1                 1                 0                 1

          2                 1                 1                 2

          3                 1                 2                 3

          4                 3                 2                 5

          5                 4                 4                 8

          6                 6                 7                 13

          7                 11               10               21

          8                 17               17               34

          9                 27               28               55

          10               45               44               89

          11               72               72               144

          Investigations continue. I have deduced that B and W are linear combinations of powers of phi, -1/phi, and the two complex cube roots of unity – therefore the period 3 difference between B and W.