%I #16 Dec 06 2019 21:44:17
%S 13,26,37,52,73,74,97,103,104,111,133,146,148,157,193,194,199,206,207,
%T 208,209,217,221,222,223,231,253,266,277,292,296,307,313,314,317,337,
%U 373,386,388,397,398,409,412,414,416,417,418,419,431,433,434,439,442,444,446,447,449,457,461,462,463,471,493,506,517,532,553,554,577,583,584,591,592,613,614,619,626,627,628,629,631,634,637,667,673,674,677,697,733,746,757,772,776,787,793,794,797,817,853
%N Dragon Curve triple point lower inverses. If D:[0,1] is a Dragon curve, then besides n, there are two larger integers p, q (with p < q) with D(A(n)/(15*2^k)) = D(A(p)/(15*2^k)) = D(A(q)/(15*2^k)), where k is any integer > log_2(A(q)/15).
%C For the triples grouped, use Dragon(A260748(n)) = Dragon(A260749(n)) = Dragon(A260750(n)). (I.e., they're "conformal".)
%C Unlike A260747, A260749, A260750, and A260482, the first differences show a fractal pattern of repetitions.
%C See dragun in the MATHEMATICA section for an exact evaluator of a continuous, spacefilling Dragon function, and undrag, its multivalued inverse.
%H Brady Haran and Don Knuth, <a href="https://www.youtube.com/watch?v=v678Em6qyzk">Wrong turn on the Dragon</a>, Numberphile video (2014)
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dragon_curve">Dragon curve</a>
%e For definiteness, we choose the Dragon in the complex plane with Dragon(0) = 0, Dragon(1) = 1, Dragon(1/3) = 1/5+2i/5
%e Then using A(1) = 13, for k=0,1,2, {dragun[13/15], dragun[13/30], dragun[13/60]}
%e -> {{2/3 - I/3}, {1/2 + I/6}, {1/6 + I/3}}
%e These have inverse images undrag/@First/@%
%e {{13/15}, {13/30, 7/10, 23/30}, {13/60, 7/20, 23/60}}
%e k=0 is too small--7/5 and 23/15 are off the end of the curve!
%e dragun[13/15/2^k] = dragun[21/15/2^k] = dragun[23/15/2^k], which empirically = (2/3 - I/3) (1/2 + I/2)^k
%t (* by Julian Ziegler Hunts *)
%t piecewiserecursivefractal[x_, f_, which_, iters_, fns_] := piecewiserecursivefractal[x, g_, which, iters, fns] = ((piecewiserecursivefractal[x, h_, which, iters, fns] := Block[{y}, y /. Solve[f[y] == h[y], y]]); Union @@ ((fns[[#]] /@ piecewiserecursivefractal[iters[[#]][x], Composition[f, fns[[#]]], which, iters, fns]) & /@ which[x]));
%t dragun[t_] := piecewiserecursivefractal[t, Identity, Piecewise[{{{1}, 0 <= # <= 1/2}, {{2}, 1/2 <= # <= 1}}, {}] &, {2*# &, 2*(1 - #) &}, {(1 + I)*#/2 &, (I - 1)*#/2 + 1 &}]
%t undrag[z_] := piecewiserecursivefractal[z, Identity, If[-(1/3) <= Re[#] <= 7/6 && -(1/3) <= Im[#] <= 2/3, {1, 2}, {}] &, {#*(1 - I) &, (1 - #)*(1 + I) &}, {#/2 &, 1 - #/2 &}]
%t DeleteDuplicates[Reap[Do[If[Length[#] > 2, Sow[15*64*#[[1]]]] &@
%t undrag[dragun[k/15/64][[1]]], {k, 0, 288*3}]][[2, 1]]]
%t (* or 128 or 256 or ...*)
%Y A260747 = A260748 U A260749 U A260750 = Superset of 3*A260482.
%K nonn,frac,obsc
%O 1,1
%A _Bill Gosper_, Jul 30 2015