[math-fun] weird continued fraction
Benoit Cloitre sent me a draft containing a recurrence identity implying a convergent matrix product: [ 2 2 ] [ --------- --------- ] [ sqrt(%pi) sqrt(%pi) ] [ ] = [ %pi 2 %pi 2 ] [ sqrt(---) - sqrt(---) sqrt(---) - sqrt(---) ] [ 2 %pi 2 %pi ] inf [ sqrt(n + 1) ] /===\ [ ------------- 1 ] | | [ n sqrt(n + 2) ] | | [ ] , | | [ sqrt(n) ] n = 1 [ ----------- 0 ] [ sqrt(n + 2) ] giving the c.f. (if you'll pardon my ascii), _ /2 /3 2 /2 / - / - v V 1 V 1 ------ = ---- + ------------------ . pi - 2 1 /3 /4 / - / - V 2 V 2 ---- + ----------- 2 /4 /5 / - / - V 3 V 3 ---- + ---- 3 . . . Nontriangular 2x2 products usually diverge, and you get the c.f. as the limit of the top row over the bottom. If we can find decent asymptotics to c.f. numerators (and denominators), we might scale them out of the matrix to produce more interesting identities of which the c.f. is a projection (which, in the case above, conceals a sqrt(pi)). --Bill Gosper P.S., note the near "recurrence" of our old friend, pi/2+2/pi, the expected magnitude of a 3D rotation, and once-believed optimum "sofa problem" area.
Benoit Cloitre's message also contained an observation linking two fairly recent math-fun threads--hypergeometric integers and square-free (stutter-free) strings. I conjectured the integerhood of a series which Neil catalogued as A087659 and Alec Mihailovs subsequently recognized as row-sums of an integer triangle of special cases of the hook length formula. (So the A087659 Comment could use updating.) Cloitre has characterized A087659 mods 2 and 3. Remarkably, A087659(9k+6) mod 3 = 2*A014578(k+1), the binary expansion of the "Thue constant", 110110111110110111110110110..., wherein the 3nth bit is the complement of the nth. It is probably my fault if this constant is misattributed. It was "computed" circa 1971 by a very simple Life pattern (as a diagonal row of blinkers), an obvious case of the (Thue-Siegel-) Roth criterion for tanscendence, since the error after 3^n bits is ~2^-3^(n+1) = O(denominator^-3). I probably should have called it Roth's constant. The "Thue-Morse", or parity constant 0110100110010110..., wherein the nth bit is the complement of the (n-2^floor(lg n-1))th, is correctly named, since (if memory serves) Thue proved its transcendence and Morse its "irrepentance" (nonrepetition). E.g., if you shift 1 bit,it's squarefree base 4. Then, if you confuse 10 and 01, it's still squarefree. Another vague memory: again around 1970, Donald Eastlake wrote PDP-6 machine code to brute-force tree-search for an "infinite" squarefree base 3, and the prgram bogged down after a few dozen bits, (mis)leading him to suspect nonexistence. What might have happened is that trying "0" first at each level of the search traps you into a sequence with no infinite continuation, but a huge number of false hopes. --rwg PS, parity constant is A010060.
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R. William Gosper