|
%I #16 Sep 05 2019 04:37:11
%S 0,0,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,1,1,
%T 0,0,1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0,0,1,0,1,1,0,1,1,0,1,1,1,1,0,
%U 0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,1,0
%N Trace(Pi), where trace is defined in Comments.
%C Suppose r > 0. Let p(n)/q(n) be a sequence of rational numbers with limit r. Let t(n) be the trace of (p(n),q(n)), and define trace(r) = lim(trace(p(n),q(n))).
%C The trace of a pair (x,y) of positive integers is defined at A228469 as a 01 sequence determined by the accelerated Euclidean algorithm; specifically, in the steps which take (x,y) to gcd(x,y), a "0" records steps where (u,v)->(v, (v mod u)), and "1" records steps where (u,v)->(v, v - (v mod u)); i.e., "0" if (v mod u) <= v/2 and "1" otherwise.
%C Conjectures: (1) trace(pi) is not periodic; (2) if k is a positive integer, then trace(sqrt(k)) is purely periodic; (3) if r is a quadratic irrational, or if r = s*e where s is a positive rational number, or if r = s*e^2 where s is a positive rational number, then trace(r) is eventually periodic.
%e The first 5 convergents to Pi are 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215. Write these as (3,1), (22,7), (333,106), (355,113), (103993,33102) and apply w to each repeatedly until reaching ( . , 1). The corresponding 01 words are 0, 00, 00, 001, 001; further convergents yield 0010, 0011, 00111, 001111, and so on, with limit as asserted.
%t $MaxExtraPrecision = Infinity; r = Pi; z = 800; p = Convergents[r, z]; pairs = Table[{Numerator[p][[k]], Denominator[p][[k]]}, {k, 1, z}]; t[{x_, y_, _}] := t[{x, y}]; t[{x_, y_}] := Prepend[If[# > y - #, {y - #, 1}, {#, 0}], y] &[Mod[x, y]]; userIn2[{x_, y_}] := Most[NestWhileList[t, {x, y}, (#[[2]] > 0) &]]; tt = Map[Map[#[[3]] &, Rest[userIn2[#]]] &, pairs]; t1 = Last[tt] (* _Peter J. C. Moses_, Aug 20 2013 *)
%Y Cf. A228469.
%K nonn
%O 0
%A _Clark Kimberling_, Aug 23 2013
|
