Suppose r>0. Let's mimic the standard algorithm for finding the simple continued fraction of r, but, instead of taking successive numerators =1, take each to be the most recently generated "a(k)" - like this: r(0)=r, a(0)=[r(0)] r(1)=a(0)/(r(0)-a(0)) a(1)=[r(1)] r(2)=a(1)/(r(1)-a(1)) a(2)=[r(2)], and so on. Call the sequence <a(0),a(1),a(2),...> the self-generating continued fraction of r, and denote it by S(r). Has anyone encountered this previously? One very attractive example has been known for a long time: 1/(e-2) = <1,2,3,4,5,6,7,8,...>. An easy problem is to determine all r for which S(r) is a constant sequence. Probably more interesting, though, are examples like these: <1,2,4,8,16,32,64,...> = 1.4086159797... <1,3,5,7,9,11,13,...> = 1.2831923416... <1,4,7,10,13,16,...> = 1.221107010123... Are these numbers "known"? Let R_n denote the set of r corresponding to the arithmetic sequences of positive integers congruent to k mod n for k=1,2,...,n. What can be said about the numbers in R_n? Best wishes, Clark Kimberling