A260750 Dragon Curve triple point upper inverses. If D:[0,1] is a Dragon curve, then if k is any integer > log_2(A(n)/15), besides n there are two smaller integers p and q with D(A(p)/(15*2^k)) = D(A(q)/(15*2^k)) = D(A(n)/(15*2^k)).

23, 46, 47, 92, 83, 94, 107, 173, 184, 163, 143, 166, 188, 167, 203, 214, 329, 346, 341, 368, 333, 227, 331, 326, 293, 283, 263, 286, 287, 332, 376, 377, 323, 334, 369, 347, 383, 406, 428, 407, 658, 659, 692, 682, 736, 671, 666, 663, 661, 443, 454, 569, 662, 652, 586, 581, 573, 467, 571, 566, 533, 523, 503, 526, 527, 572, 563, 574, 587, 653, 664, 643, 752, 623, 754, 753, 646, 751, 668, 739, 761, 738, 647, 737, 683, 694, 729, 707, 743, 766, 767, 812, 856, 857, 803, 814, 849, 827, 863

Offset

1,1

Comments

See dragun in the MATHEMATICA section for an exact evaluator of a continuous, spacefilling Dragon function, and undrag, its multivalued inverse.

For the triples grouped, use Dragon(A260748(n)) = Dragon(A260749(n)) = Dragon(A260750(n)). (I.e., they're "conformal".)

Links

Table of n, a(n) for n=1..99.

Brady Haran and Don Knuth, Wrong turn on the Dragon, Numberphile video (2014)

Wikipedia, Dragon curve

Example

For definiteness, we choose the Dragon in the complex plane with Dragon(0) = 0, Dragon(1) = 1, Dragon(1/3) = 1/5+2i/5
(I^2:=-1)  Then using A(3) = 47, for k=2,3,4, {dragun[47/60], dragun[47/120],dragun[47/240]}
-> {{2/3 + I/6}, {1/4 + (5 I)/12}, {-(1/12) + I/3}}
These have inverse images undrag/@First/@%
{{37/60, 13/20, 47/60}, {37/120, 13/40, 47/120}, {37/240, 13/80, 47/240}}
dragun[47/15/2^k] = dragun[39/15/2^k] = dragun[37/15/2^k], which empirically = (5/3 - I) (1 + I)^k 2^(-1 - k)
so every eighth point is 5/6-I/2 over a power of 16.

Mathematica

(* by Julian Ziegler Hunts *)
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]));
dragun[t_] := piecewiserecursivefractal[t, Identity, Piecewise[{{{1}, 0 <= # <= 1/2}, {{2}, 1/2 <= # <= 1}}, {}] &, {2*# &, 2*(1 - #) &}, {(1 + I)*#/2 &, (I - 1)*#/2 + 1 &}]
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 &}]
DeleteDuplicates[Reap[Do[If[Length[#] > 2, Sow[15*64*#[[3]]]] &@
     undrag[dragun[k/15/64][[1]]], {k, 0, 288*3}]][[2, 1]]]
(* or 128 or 256 or ... *)

Crossrefs

A260747 = A260748 U A260749 U A260750 = Superset of 3*A260482.

Sequence in context: A081329 A323046 A025022 * A094694 A164010 A132682

Adjacent sequences: A260747 A260748 A260749 * A260751 A260752 A260753

Keyword

nonn

Author

Bill Gosper, Jul 30 2015

Status

approved