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A328078
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Number of regions after n generations of Jim Conant's iterative dissection of a square.
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33
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1, 2, 3, 5, 9, 15, 27, 48, 91, 169, 325, 618, 1201, 2319, 4527, 8804, 17227, 33649, 65929, 129046, 252997, 495779, 972339, 1906520, 3739775, 7335029, 14389629, 28227578, 55378713, 108642983, 213148903, 418176700, 820441299, 1609656953, 3158089841, 6196050718
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refs;
listen;
history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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In view of the comment below from Robert Fathauer that this is a generalization of the Sierpinski Gasket, I propose that this structure be called the Conant Gasket. - N. J. A. Sloane, Oct 29 2019
The following description of the dissection is based on an email message from Robert Fathauer. Start with a unit square.
1. Draw a vertical line from bottom to top, dividing the square in half.
2. Go to the left edge, and halfway up draw a line from left to right. Stop when you reach the vertical line.
3. Return to the bottom edge and draw vertical lines at the midpoints of the remaining intervals (at the 1/4 and 3/4 points for the second pass), from bottom to top. If you reach a line, stop, skip to the next line and start again, stopping the next time you reach a line.
4. Return to the left edge and draw horizontal lines at the midpoints of the remaining intervals, from left to right. If you reach a line, stop, skip to the next line and start again, stopping the next time you reach a line.
Repeat steps 3 and 4.
Might be called a basket-weave dissection, because the lines go alternately over and under the perpendicular lines, like the warp and woof on a loom. - N. J. A. Sloane, Oct 16 2019
Robert Fathauer and Rémy Sigrist independently observed that the sizes of the regions along the bottom edge at even generations appear to be converging to a sequence (it is now A330569) closely related to the 2-adic valuation of the column number (see illustration). A similar thing happens along the left edge. - N. J. A. Sloane, Oct 14 2019 and Jan 07 2020 [Thanks to M. Douglas McIlroy for pointing out an error in an earlier version of this comment.]
Conjecture: The left half of the dissection at generation n is essentially the same as the whole of the dissection at generation n-1. To see this, take the dissection at generation n-1, rotate it counterclockwise by 90 degrees, then reflect it about a vertical line through its center, and compress it in the horizontal direction by a factor of 2. The result is essentially the left half of generation n. - N. J. A. Sloane, Oct 14 2019
This can be regarded as a generalization of the Sierpinski Gasket. For if you start with an equilateral triangle, draw a line from a midpoint to another midpoint, rotate 120 degrees, repeat, do that again, then draw the 1/4 and 3/4 lines, etc., using the over-and-under interlacing, you end up with the Sierpinski Gasket, as in A047999 (see illustration here for first steps). - Robert Fathauer, Oct 16 2019
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REFERENCES
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Jim Conant, Posting to the "Bridges - Art and Mathematics" Facebook page, Oct 05 2019. [The URL is said to be https://www.facebook.com/groups/20666497429/, but I was unable to access it.]
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LINKS
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FORMULA
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G.f.: (1 - 3*x^2 + x^3 - 2*x^4 - 5*x^5 + 8*x^6 - x^7 - 4*x^8 + 8*x^9 - 4*x^10 + 4*x^12) / ((1 - x)*(1 + x^2)*(1 - 2*x^2)*(1 - x - 2*x^2 + 2*x^3 - 4*x^4 + 4*x^6 - 4*x^7)).
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 7*a(n-4) - 4*a(n-5) - 12*a(n-6) + 20*a(n-7) - 12*a(n-8) + 12*a(n-10) - 16*a(n-11) + 8*a(n-12) for n>12.
(End)
Gfun still gives the same recurrence and g.f. using all 40 terms.
The seven roots of the denominator polynomial are
-0.871341341681075,
-0.661031992215005,
0.0876691186562792 - 0.808024853721450 I,
0.0876691186562792 + 0.808024853721450 I,
0.509688436780776,
0.923673329901373 - 0.660261157442008 I,
0.923673329901373 + 0.660261157442008 I,
and the magnitudes of the complex roots are:
0.0876691186562792^2 + 0.808024853721450^2 = 0.6605900386; sqrt = 0.8127669030
0.923673329901373^2 + 0.660261157442008^2 = 1.289117216; sqrt = 1.135392979
(End)
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MATHEMATICA
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(*Code written by Jim Conant, Oct 12 2019*)
n = 4;
size = 2^n;
V = Table[0, {size + 1}, {size + 1}];
H = Table[0, {size + 1}, {size + 1}];
Flag = vert;
vcol = 2^(n - 1);
hcol = 2^(n - 1);
down = 1;
up = 2;
(*************************************************)
While[hcol > .9,
If[Flag == vert, Flag = horiz;
Do[Pen = down;
Do[If[Pen == up, If[H[[i, j]] != 0, Pen = down], V[[i, j]] = 1/n;
If[H[[i, j]] != 0, Pen = up]], {j, 1, size, 1}], {i, vcol,
size - 1, 2*vcol}];
vcol = vcol/2];
If[Flag == horiz, Flag = vert;
Do[Pen = down;
Do[If[Pen == up, If[V[[i, j]] != 0, Pen = down], H[[i, j]] = 1/n;
If[V[[i, j]] != 0, Pen = up]], {i, 1, size, 1}], {j, hcol,
size - 1, 2*hcol}];
hcol = hcol/2];
n = n - 1];
(**Display graphics with data from V and H********)
G = {};
Do[Do[If[V[[i, j]] != 0, G = Append[G, Line[{{i, j - 1}, {i, j}}]]];
If[H[[i, j]] != 0, G = Append[G, Line[{{i - 1, j}, {i, j}}]]], {i,
size}], {j, size}];
G = Join[G, {Line[{{0, 0}, {0, size}}], Line[{{0, 0}, {size, 0}}],
Line[{{size, 0}, {size, size}}], Line[{{0, size}, {size, size}}]}];
Show[Graphics[G], AspectRatio -> Automatic]
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PROG
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(C#) See Links section.
(C++) See Links section.
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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