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A328078 Number of regions after n generations of Jim Conant's iterative dissection of a square.

%I A328078 #187 Jan 21 2023 08:16:02
%S A328078 1,2,3,5,9,15,27,48,91,169,325,618,1201,2319,4527,8804,17227,33649,
%T A328078 65929,129046,252997,495779,972339,1906520,3739775,7335029,14389629,
%U A328078 28227578,55378713,108642983,213148903,418176700,820441299,1609656953,3158089841,6196050718
%N A328078 Number of regions after n generations of Jim Conant's iterative dissection of a square.
%C A328078 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
%C A328078 The following description of the dissection is based on an email message from _Robert Fathauer_. Start with a unit square.
%C A328078 1. Draw a vertical line from bottom to top, dividing the square in half.
%C A328078 2. Go to the left edge, and halfway up draw a line from left to right. Stop when you reach the vertical line.
%C A328078 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.
%C A328078 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.
%C A328078 Repeat steps 3 and 4.
%C A328078 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
%C A328078 _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.]
%C A328078 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
%C A328078 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
%D A328078 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.]
%H A328078 Rémy Sigrist, <a href="/A328078/b328078.txt">Table of n, a(n) for n = 0..42</a>
%H A328078 Jim Conant, <a href="/A328078/a328078_1.png">Illustration for a(4) = 9.</a> [Produced by the Mma program with n=2.]
%H A328078 Jim Conant, <a href="/A328078/a328078.png">Illustration for a(8) = 91.</a> [Produced by the Mma program with n=4.]
%H A328078 Jim Conant, <a href="/A328078/a328078_2.png">Illustration for a(10) = 325.</a> [Produced by the Mma program with n=5.]
%H A328078 Robert Fathauer, <a href="/A328078/a328078.jpg">Colored illustration for a(16).</a> [The same feature is colored the same at different scales to elucidate the fractal nature of the tiling.]
%H A328078 Robert Fathauer, <a href="/A328078/a328078_4.png">The analogous construction starting from a triangle leads to the Sierpinski Gasket</a>
%H A328078 Douglas McIlroy, <a href="https://digitalcommons.dartmouth.edu/cs_tr/385/">Reasoning about the Conant Gasket</a>, Dartmouth Computer Science Technical Report TR2023-1003. https://digitalcommons.dartmouth.edu/cs_tr/385, 2023
%H A328078 Douglas McIlroy, <a href="/A328078/a328078_6.pdf">Reasoning about the Conant Gasket</a>, local copy, with permission.
%H A328078 Rémy Sigrist, <a href="/A328078/a328078_3.png">The sizes of the regions along the bottom edge at even generations appear to be given by sequence A330569</a>
%H A328078 Rémy Sigrist, <a href="/A328078/a328078.txt">C# program for A328078</a>
%H A328078 Rémy Sigrist, <a href="/A328078/a328078_1.txt">C++ program for A328078</a>
%H A328078 Rémy Sigrist, <a href="/A328078/a328078_3.txt">C++ program for A328078</a> [with moderate memory use]
%H A328078 N. J. A. Sloane, <a href="/A328078/a328078_2.txt">Notes on the Conant Gasket, the Conant Lattice, and Associated Sequences</a>, Preliminary version, Aug 23 2020
%H A328078 N. J. A. Sloane, <a href="/A328078/a328078_5.pdf">The Even Conant Lattice</a>
%H A328078 N. J. A. Sloane, <a href="/A328078/a328078_4.pdf">The Odd Conant Lattice</a>
%H A328078 N. J. A. Sloane, <a href="/A328078/a328078_1.pdf">Illustrations for a(1) though a(7).</a>
%H A328078 N. J. A. Sloane, <a href="/A328078/a328078.pdf">Colored illustration for a(8) = 91, showing successive stages in the construction.</a> [Thin black lines were used to construct generations 0 through 4, red lines produce generation 5, green = 6, thick black = 7, thick blue = 8.]
%H A328078 N. J. A. Sloane, <a href="https://vimeo.com/457349959">Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows</a>, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)
%H A328078 (Unknown author), <a href="/A328078/a328078_2.pdf">A similar-looking fractal</a> If anyone can identify this fractal, please let me know! - _N. J. A. Sloane_, Jan 14 2020
%F A328078 Conjectures from _Colin Barker_, Oct 15 2019: (Start)
%F A328078 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)).
%F A328078 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.
%F A328078 (End)
%F A328078 Comment from _N. J. A. Sloane_, Sep 08 2020 (Start)
%F A328078 Gfun still gives the same recurrence and g.f. using all 40 terms.
%F A328078 The seven roots of the denominator polynomial are
%F A328078 -0.871341341681075,
%F A328078 -0.661031992215005,
%F A328078 0.0876691186562792 - 0.808024853721450 I,
%F A328078 0.0876691186562792 + 0.808024853721450 I,
%F A328078 0.509688436780776,
%F A328078 0.923673329901373 - 0.660261157442008 I,
%F A328078 0.923673329901373 + 0.660261157442008 I,
%F A328078 and the magnitudes of the complex roots are:
%F A328078 0.0876691186562792^2 + 0.808024853721450^2 = 0.6605900386; sqrt = 0.8127669030
%F A328078 0.923673329901373^2 + 0.660261157442008^2 = 1.289117216; sqrt = 1.135392979
%F A328078 (End)
%t A328078 (*Code written by Jim Conant, Oct 12 2019*)
%t A328078 n = 4;
%t A328078 size = 2^n;
%t A328078 V = Table[0, {size + 1}, {size + 1}];
%t A328078 H = Table[0, {size + 1}, {size + 1}];
%t A328078 Flag = vert;
%t A328078 vcol = 2^(n - 1);
%t A328078 hcol = 2^(n - 1);
%t A328078 down = 1;
%t A328078 up = 2;
%t A328078 (*************************************************)
%t A328078 While[hcol > .9,
%t A328078   If[Flag == vert, Flag = horiz;
%t A328078    Do[Pen = down;
%t A328078     Do[If[Pen == up, If[H[[i, j]] != 0, Pen = down], V[[i, j]] = 1/n;
%t A328078       If[H[[i, j]] != 0, Pen = up]], {j, 1, size, 1}], {i, vcol,
%t A328078      size - 1, 2*vcol}];
%t A328078    vcol = vcol/2];
%t A328078   If[Flag == horiz, Flag = vert;
%t A328078    Do[Pen = down;
%t A328078     Do[If[Pen == up, If[V[[i, j]] != 0, Pen = down], H[[i, j]] = 1/n;
%t A328078       If[V[[i, j]] != 0, Pen = up]], {i, 1, size, 1}], {j, hcol,
%t A328078      size - 1, 2*hcol}];
%t A328078    hcol = hcol/2];
%t A328078   n = n - 1];
%t A328078 (**Display graphics with data from V and H********)
%t A328078 G = {};
%t A328078 Do[Do[If[V[[i, j]] != 0, G = Append[G, Line[{{i, j - 1}, {i, j}}]]];
%t A328078    If[H[[i, j]] != 0, G = Append[G, Line[{{i - 1, j}, {i, j}}]]], {i,
%t A328078     size}], {j, size}];
%t A328078 G = Join[G, {Line[{{0, 0}, {0, size}}], Line[{{0, 0}, {size, 0}}],
%t A328078     Line[{{size, 0}, {size, size}}], Line[{{0, size}, {size, size}}]}];
%t A328078 Show[Graphics[G], AspectRatio -> Automatic]
%o A328078 (C#) See Links section.
%o A328078 (C++) See Links section.
%Y A328078 Cf. A001511, A328079 (first differences), A328080 (a bisection), A328081 (numbers of regions of each size), A337642 (coordination sequence), A337780 (length of drawn lines).
%Y A328078 Cf. also A047999, A330569.
%K A328078 nonn,nice
%O A328078 0,2
%A A328078 _N. J. A. Sloane_, Oct 13 2019, based on emails from _Robert Fathauer_
%E A328078 a(9)-a(26) from _Rémy Sigrist_, Oct 14 2019
%E A328078 a(27)-a(33) from _Rémy Sigrist_, Oct 15 2019
%E A328078 a(34)-a(35) added by _N. J. A. Sloane_, Sep 06 2020 using _Rémy Sigrist_'s C++ program.