Reiterating, for a burst of terms L := {a, b, . . .}, the Gauss-Kuzmin distribution predicts a probability of cfprob[L_List] := Abs@Log[2, (1 + 1/FromContinuedFraction@L)/(1 + 1/FromContinuedFraction@MapAt[# + 1 &, L, -1])] or cfprob[r : (Integer_ | Rational_)] := cfprob@ContinuedFraction@r where r := a+1/(b + 1/. . . ) I neglected to compare the probability of . . . /(1+1/(2 + 1/(3 +. . .))) cfprob@{1, 2, 3} Log[221/220]/Log[2] N@% 0.00654284586677209 against the similar but incorrect cfprob /@ Range@3 {Log[4/3]/Log[2], Log[9/8]/Log[2], Log[16/15]/Log[2]} Times @@ % // N 0.00656656380451684 assuming that those three probabilities were independent. Since Log[4/3]/Log[2] ~ 41% of π's terms ought to be 1, let's find in π some bursts of at least a dozen non-1s: Function[cf, Take[cf, #] & /@ Echo@SequencePosition[cf, {Repeated[Except@1, {12, \[Infinity]}]}, Overlaps -> False]]@ContinuedFraction[\[Pi], 9999] // tim {{3191,3202}, {3315,3326}, {6778,6790}} During evaluation of In[136]:= 1058.655331 (*secs, almost prohibitive *), 3 {{2, 4, 2, 2, 32, 2, 4, 16, 3, 46, 2, 5}, {2, 33, 3, 3, 2, 3, 9, 6, 4, 4, 5, 7}, {32, 2, 2, 9, 3, 2, 3, 2, 5, 3, 3, 2, 54}} The same computation extended to the bursts ≥13 of non-1s among 99999 terms has thus far consumed most of a day. (Veridically prohibitive.) (Small scale experiments just now suggest that the time complexity is *worse* than cubic! Time to back off. 30000 should run overnight.) —rwg