An unusual continued fraction

Abstract

We consider the real number σ \sigma with continued fraction expansion [ a 0 , a 1 , a 2 , … ] = [ 1 , 2 , 1 , 4 , 1 , 2 , 1 , 8 , 1 , 2 , 1 , 4 , 1 , 2 , 1 , 16 , … ] [a_0, a_1, a_2,\ldots ] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots ] , where a i a_i is the largest power of 2 2 dividing i + 1 i+1 . We show that the irrationality measure of σ 2 \sigma ^2 is at least 8 / 3 8/3 . We also show that certain partial quotients of σ 2 \sigma ^2 grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.