MathDB

Denote

\phi=\frac{\sqrt{5}+1}{2}

and consider the set of all finite binary strings without leading zeroes. Each string

S

has a “base-

\phi

” value

p(S)

. For example,

p(1101)=\phi^3+\phi^2+1

. For any positive integer n, let

f(n)

be the number of such strings S that satisfy

p(S) =\frac{\phi^{48n}-1}{\phi^{48}-1}

. The sequence of fractions

\frac{f(n+1)}{f(n)}

approaches a real number

c

n

goes to infinity. Determine the value of

c

Problem Statement