weave[]ing continued fraction expansions on the unit square

[Joseph Biberstine]

All,

	Let weave[] (a function surely already well-studied) take two real
numbers on (0,1) and return the real number given by interpreting {the
sequence made by interleaving their respective continued fraction
expansions} as a continued fraction expansion.  Fix all expansions to be
in the usual strict form (none ending in 1, all partial quotients
positive beyond index 0).  In the following, all "0;" beginning the c.f.
expansion will be implied.

weave[1/2, 1/3] = weave[{2}, {3}] = {2,3} = 3/7
weave[{1,2,3}, {11,12,13,14,15}] = {1,11,2,12,3,13,14,15}
weave[{1,2,3,4,5}, {11,12}] = {1,11,2,12,3,4,5}
etc.

	I implement it sloppily in Mathematica as:

weave[cfA_, cfB_] := (listA = Rest[ContinuedFraction[[cfA]]]; listB =
Rest[ContinuedFraction[Rationalize[cfB]]]; diff = Length[listA] -
Length[listB]; Which[diff > 0, res = {listA, Join[listB, Table[{},
{index, 1, diff}]]}, diff < 0, res = {Join[listA, Table[{}, {index, 1,
-diff}]], listB}, True, res = {listA, listB}];
FromContinuedFraction[Join[{0}, Flatten[Transpose[res]]]]);

	Note the use of Rationalize[] to iron out some unfortunate "features"
of ContinuedFraction[].

	Let c = Integrate[weave[a,b], {a,0,1}, {b,0,1}].  What is c?  My
machine calculates the estimate c = 0.469447 in six minutes using all
default precisions and accuracies in Mathematica with the following code:

dx = .001;
Sum[weave[a, b]*dx^2, {b, 10^-8, 1 - 10^-8, dx}, {a, 10^-8, 1 - 10^-8, dx}]

	This estimate probably isn't accurate even to two decimals.  What is c
and what existing literature discusses the properties of weave[]?
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