

A260747


Consolidated Dragon Curve triple points. If D:[0,1] is a Dragon curve, then besides n, there are two other integers p and 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(max(A(n),A(p),A(q))/15).


6



13, 21, 23, 26, 37, 39, 42, 46, 47, 52, 73, 74, 78, 81, 83, 84, 92, 94, 97, 99, 103, 104, 107, 111, 113, 133, 141, 143, 146, 148, 156, 157, 159, 162, 163, 166, 167, 168, 171, 173, 184, 188, 193, 194, 198, 199, 201, 203, 206, 207, 208, 209, 211, 213, 214, 217, 219, 221, 222, 223, 226, 227, 231, 233, 253, 261, 263, 266, 277, 279, 282, 283, 286, 287
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OFFSET

1,1


COMMENTS

It appears that every Dragon triple point is an image of A(n)/(15*2^k) for three different n and some k.
For the triples grouped, use
Dragon(A260748(n)) = Dragon(A260749(n)) = Dragon(A260750(n)). (I.e., they're "conformal".)
The first differences of this sequence appear to comprise only 1, 2, 3, 4, 5, 8, 11, 20, and 21. 21 occurs only twice for A(n) < 30720.
See dragun in the MATHEMATICA section for an exact evaluator of a continuous, spacefilling Dragon function, and undrag, its multivalued inverse.
Even excluding multiples of 5, it is NOT the case that A260747 contains 7*A260747, e.g., 7*13=91 is missing.


LINKS

Table of n, a(n) for n=1..74.
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
Then using A(1) = 13, for k=0,1,2, {dragun[13/15], dragun[13/30], dragun[13/60]}
> {{2/3  I/3}, {1/2 + I/6}, {1/6 + I/3}} (where I^2:=1)
These have inverse images undrag/@First/@%
{{13/15}, {13/30, 7/10, 23/30}, {13/60, 7/20, 23/60}}
k=0 is too small7/5 and 23/15 are off the end of the curve!
dragun[13/15/2^k] = dragun[7/5/2^k] = dragun[23/15/2^k], which empirically = (2/3  I/3) (1/2 + I/2)^k


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 &}]
Reap[Do[If[Length[undrag[dragun[k/15/32][[1]]]] > 2, Sow[k]], {k, 0, 288}]][[2, 1]]


CROSSREFS

A260747 = A260748 U A260749 U A260750 = Superset of 3*A260482.
Sequence in context: A066515 A166656 A230498 * A032693 A049745 A332512
Adjacent sequences: A260744 A260745 A260746 * A260748 A260749 A260750


KEYWORD

nonn,frac,obsc


AUTHOR

Bill Gosper, Jul 30 2015


EXTENSIONS

Corrected subtle bug in NAME section, plus three tweaks to EXAMPLE. Tweaked comment.  Bill Gosper, Jul 31 2015


STATUS

approved



