|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
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
|
|
|
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 small--7/5 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
|
|
|
KEYWORD
|
nonn,frac,obsc
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Corrected subtle bug in NAME section, plus three tweaks to EXAMPLE. Tweaked comment. - Bill Gosper, Jul 31 2015
|
|
STATUS
|
approved
|
|
|
|