Here is a special case of theorem 1 of Harald Niederreiter: Dyadic fractions with small partial quotients, Monatshefte f"ur Mathematik 101,4 (December 1986) 309-315 Let p[0]=1, p[1]=3, p[2]=7, p[3]=113 and if n>=3 let p[n+1]=2^(2^n)*p[n]-1. (This sequence is not in, but should be in, the oeis.) Then the rational number p[n]/2^(2^n) has continued fraction with all partial quotients either 1, 2, or 3. For n large this rational number approaches 0.441390990978106856291920262469474778296688624079447195658789... 1/2 = [0; 2] 3/4 = [0; 1, 3] 7/16 = [0; 2, 3, 2] 113/256 = (7*16+1)/2^8 = [0; 2, 3, 1, 3, 3, 2] 28927/65536 = (113*256-1)/2^16 = [0; 2, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 2] 1895759871/2^32 = (28927*2^16-1)/2^32 = [0; 2, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 2] 8142226647014178815/2^64 = (1895759871*2^32-1)/2^64 = [0; 2, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 2] 150197571147608796277790585648096215039/2^128 = (8142226647014178815*2^64-1)/2^128 = [0; 2, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 2] 51109585015884376828428305273936708056145924221155280299477400051792199286783/2^256 = (150197571147608796277790585648096215039*2^128-1)/2^256 = [0; 2, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 1, 3, 2] -- Warren D. Smith