login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A260749 Dragon Curve triple point middle inverses. If D:[0,1] is a Dragon curve,  then besides n, there are two other integers p and q (with p < n < q) with D(A(p)/(15*2^k)) = D(A(n)/(15*2^k)) = D(A(q)/(15*2^k)), where k is any integer > lg(A(q)/15). 5
21, 42, 39, 84, 81, 78, 99, 171, 168, 113, 141, 162, 156, 159, 201, 198, 213, 342, 211, 336, 319, 219, 327, 226, 291, 233, 261, 282, 279, 324, 312, 309, 321, 318, 367, 339, 381, 402, 396, 399, 426, 423, 684, 422, 672, 421, 638, 649, 657, 441, 438, 453, 654, 452, 582, 451, 559, 459, 567, 466, 531, 473, 501, 522, 519, 564, 561, 558, 579, 651, 648, 593, 624, 621, 618, 749, 642, 641, 636, 633, 747, 734, 639, 669, 681, 678, 727, 699, 741, 762, 759, 804, 792, 789, 801, 798, 847, 819, 861 (list; graph; refs; listen; history; text; internal format)
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

Then using A(1) = 21, for k=1,2,3, {dragun[21/30], dragun[21/60], dragun[21/120]}

-> {{1/2 + I/6}, {1/6 + I/3}, {-1/12 + I/4}}

These have inverse images undrag/@First/@%

{{13/30, 7/10, 23/30}, {13/60, 7/20, 23/60}, {13/120, 7/40, 23/120}}

dragun[21/15/2^k] = dragun[13/15/2^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 &}]

DeleteDuplicates[Reap[Do[If[Length[#] > 2, Sow[15*64*#[[2]]]] &@

     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: A123842 A247387 A120772 * A040420 A192130 A258188

Adjacent sequences:  A260746 A260747 A260748 * A260750 A260751 A260752

KEYWORD

nonn

AUTHOR

Bill Gosper, Jul 30 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 14 06:53 EDT 2019. Contains 327995 sequences. (Running on oeis4.)