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

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

Offset: 0

Author

N. J. A. Sloane, Oct 13 2019, based on emails from Robert Fathauer

Keywords

• nonn
• nice

Comments

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

References

• 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.]

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..42
• Jim Conant, Illustration for a(4) = 9. [Produced by the Mma program with n=2.]
• Jim Conant, Illustration for a(8) = 91. [Produced by the Mma program with n=4.]
• Jim Conant, Illustration for a(10) = 325. [Produced by the Mma program with n=5.]
• Robert Fathauer, Colored illustration for a(16). [The same feature is colored the same at different scales to elucidate the fractal nature of the tiling.]
• Robert Fathauer, The analogous construction starting from a triangle leads to the Sierpinski Gasket
• Douglas McIlroy, Reasoning about the Conant Gasket, Dartmouth Computer Science Technical Report TR2023-1003. https://digitalcommons.dartmouth.edu/cs_tr/385, 2023
• Douglas McIlroy, Reasoning about the Conant Gasket, local copy, with permission.
• Rémy Sigrist, The sizes of the regions along the bottom edge at even generations appear to be given by sequence A330569
• Rémy Sigrist, C# program for A328078
• Rémy Sigrist, C++ program for A328078
• Rémy Sigrist, C++ program for A328078 [with moderate memory use]
• N. J. A. Sloane, Notes on the Conant Gasket, the Conant Lattice, and Associated Sequences, Preliminary version, Aug 23 2020
• N. J. A. Sloane, The Even Conant Lattice
• N. J. A. Sloane, The Odd Conant Lattice
• N. J. A. Sloane, Illustrations for a(1) though a(7).
• N. J. A. Sloane, Colored illustration for a(8) = 91, showing successive stages in the construction. [Thin black lines were used to construct generations 0 through 4, red lines produce generation 5, green = 6, thick black = 7, thick blue = 8.]
• N. J. A. Sloane, Conant's Gasket, Recamán Variations, the Enots Wolley Sequence, and Stained Glass Windows, Experimental Math Seminar, Rutgers University, Sep 10 2020 (video of Zoom talk)
• (Unknown author), A similar-looking fractal If anyone can identify this fractal, please let me know! - _N. J. A. Sloane_, Jan 14 2020

Crossrefs

Cf. A001511, A328079 (first differences), A328080 (a bisection), A328081 (numbers of regions of each size), A337642 (coordination sequence), A337780 (length of drawn lines).

Cf. also A047999, A330569.

Programs

Mathematica

(*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]

Formula

Conjectures from Colin Barker, Oct 15 2019: (Start)

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)

Comment from N. J. A. Sloane, Sep 08 2020 (Start)

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)

Extensions

a(9)-a(26) from Rémy Sigrist, Oct 14 2019

a(27)-a(33) from Rémy Sigrist, Oct 15 2019

a(34)-a(35) added by N. J. A. Sloane, Sep 06 2020 using Rémy Sigrist's C++ program.