As some guy recently mentioned, regular, or "floor type" cfs of square roots of rationals are periodic palindromes, e.g., if r13:=sqrt 13, r14:=sqrt 14, 1 r13 = 3 + (r13-3 = -------------------------------------------------------------), r13-1 1 1 + (----- = ------------------------------------------------) 4 r13-2 1 1 + (----- = -----------------------------------) 3 r13-1 1 1 + (----- = ---------------------) 3 r13-3 1 1 + (----- = -------) 4 3 + r13 so sqrt 13 = 3 [1 1 1 1 6]^oo, and 1 r14 = 3 + (r14 - 3 = -------------------------------------------------------), r14 - 2 1 1 + (------- = ---------------------------------------) 5 r14 - 2 1 2 + (------- = -----------------------) 2 r14 - 3 1 1 + (------- = -------) 5 r14 + 3 so sqrt 14 = 3 [1 2 1 6]^oo. "Ceiling type", or "negular" cfs have numerators of -1 instead of 1: 1 sqrt 14 = 4 + (sqrt 14 - 4 = - ---------------------------------), sqrt 14 - 4 1 4 + (----------- = - ------------) 2 sqrt 14 + 4 so sqrt 14 = 4 {4 8}^oo, as a negular expression. But it is unusual for the negular period to be shorter than the regular one. More typically, 1 r13 = 4 + (r13 - 4 = - ---------------------------------------------------------------), r13 - 5 1 3 + (------- = - ---------------------------------------------) 3 r13 - 7 1 3 + (------- = - ---------------------------) 4 r13 - 11 12 2 + (-------- = - --------) 9 r13 + 11 r13 + 11 r13 - 13 1 -------- = 2 + (-------- = - ------------------------------------------------------------), 12 12 r13 - 13 1 2 + (-------- = - ------------------------------------------) 13 r13 - 11 1 2 + (-------- = - ------------------------) 12 r13 - 7 4 2 + (------- = - -------) 9 r13 + 7 r13 + 7 r13 - 5 1 ------- = 3 + (------- = - -------------------------------------------), 4 4 r13 - 4 1 3 + (------- = - -------------------------) 3 3 8 + (r13 - 4 = - -------) r13 + 4 so sqrt(13) = 4 {3 3 2 2 2 2 2 3 3 8}^oo. The long bursts of 2s are "wannabe 1s", precluded by the ceiling rule. We can restore 1 to the inventory of partial quotients by using numerators of -1/2 instead of -1. But this actually lengthens the period while exactly preserving the per-period information density, thus reducing it per term. Note, however, that both flavors of negular cf preserve the palindromic structure of the periods. How much shorter than a regular cf can be the period of a negular cf of the same sqrt? Crudely checking for n<36353 found regular - negular periods = 48 - 30 for n = 32148, suggesting that the question should be rephrased as some sort of ratio. The greatest odd difference in that interval was 34-23 for n = 32895 For *regular* cfs of uniformly distributed reals, P(tail<x) = lg 1+x, so about 41.5% = lg 2 - lg 3/2 of the terms are 1, yet, surprisingly, the expected value of a term (partial quotient) is infinity. But then, surprise again, the *geometric* mean of the terms of these reals is the finite constant ~2.685, named for Khinchin. This lg 1+x distribution also predicts an information density of pi^2/(6 ln 2 ln 10) ~ 1.0306 digits/term, so decimal is not such an arbitrary radix after all! Gene Salamin once computed the variance of this information density to be a constant involving zeta(3), although I once saw this disputed on a Web page I didn't undertand. Has anyone done the analogous analysis for negular cfs? In particular, do those long bursts of 2s, of expected (by me) infinite expected length, manage to render the expected term finite? Finally, all these questions repeat for the "round type" cfs, which mix numerators of 1 and -1 (or 1/2 and -1/2), the former having generally shorter periods and even higher information density than regular cfs. E.g., 1 r13 = 4 + (r13 - 4 = - ---------------------------------------), r13 - 5 1 3 + (------- = - ----------------------) 3 r13 - 3 1 2 + (------- = -------) 4 r13 + 3 so sqrt 13 = 4- (3- 2 7-)^oo (note loss of palindrome and max = 2*first), and by mere coincidence, sqrt 14 = 4- (4- 8-)^oo = 4 {4 8}^oo. --rwg PS, Macsyma Inc, seems to have revived sales of version 2.4, $500 a pop on E-Bay. They even give a bugmail eadress (which I could single-handedly inundate.)