A335703 Number of regions after n generations of symmetrized Conant Gasket.

1, 2, 3, 5, 9, 16, 30, 52, 95, 174, 326, 618, 1195, 2300, 4478, 8764, 17251, 34038, 67326, 133858, 266331, 530876, 1058146, 2112904, 4216075, 8417830, 16808282, 33592910, 67130923, 134211556, 268281390, 536403988, 1072424939, 2144322638, 4287655814, 8574721850

Offset: 0

Author

Rémy Sigrist and N. J. A. Sloane, Aug 25 2020

Keywords

• nonn

Comments

Generation 0. Start with a square of side 1

Generation 1. Go to bottom edge of the square, and at the point 1/2 draw a vertical line from bottom to top.

Generation 2. Go to the left edge, and at the point 1/2 draw a line from left to right. Stop when you reach the vertical line.

Generation 3. Go to the top edge and draw downward vertical lines at the 1/4 and 3/4 points. If you reach a horizontal line, lift the pen, skip to the next horizontal line and lower the pen. Repeat until reaching the bottom edge.

Generation 4. Go to the right edge and draw horizontal lines at the 1/4 and 3/4 points. If you reach a vertical line, lift the pen, skip to the next vertical line and lower the pen. Repeat until reaching the left edge.

Generation 5. Return to the bottom edge and draw vertical lines at the 1/8, 3/8, 5/8, and 7/8 points. If you reach a horizontal line, lift the pen, skip to the next horizontal line and lower the pen. Repeat until reaching the top edge.

Generation 6. Return to the left edge and draw horizontal lines at the points 1/8, 3/8, 5/8, 7/8, following the same rules.

Continue in this way, visiting each side in turn.

At generations 2k-1 and 2k, the lines start at the points 1/2^k, 3/2^k, 5/2^k, ..., (2^k-1)/2^k.

The tick marks on the edges of the illustrations indicate which side of the square the lines start from.

This is similar to Conant's Gasket as described in A328078, except there lines were drawn alternately from the bottom and left edges of the square, which led to a figure with a strong bias to the North-East (see Robert Fathauer's colored illustration in A328078 of the gasket after 16 generations).

The present definition produces a more homogeneous design, although now there is no obvious fractal structure.

Links

• Rémy Sigrist, Illustration for a(0)
• Rémy Sigrist, Illustration for a(1)
• Rémy Sigrist, Illustration for a(2)
• Rémy Sigrist, Illustration for a(3)
• Rémy Sigrist, Illustration for a(4)
• Rémy Sigrist, Illustration for a(5)
• Rémy Sigrist, Illustration for a(6)
• Rémy Sigrist, Illustration for a(7)
• Rémy Sigrist, Illustration for a(8)
• Rémy Sigrist, Illustration for a(9)
• Rémy Sigrist, Illustration for a(10)
• Rémy Sigrist, Illustration for a(11)
• Rémy Sigrist, Illustration for a(12)
• Rémy Sigrist, Illustration for a(13)
• Rémy Sigrist, Illustration for a(14)
• Rémy Sigrist, Illustration for a(15)
• Rémy Sigrist, Illustration for a(16)
• Rémy Sigrist, Illustration for a(18)
• Rémy Sigrist, Illustration of generations 0 to 18 (Animated gif)
• Rémy Sigrist, C++ program for A335703
• N. J. A. Sloane, Illustration of generations 0 through 6.
• N. J. A. Sloane, Illustration of generations 7 (black lines) and 8 (both black and red lines).
• 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)

Crossrefs

Cf. A328078, A328080, A139250, A160124.

For bisections see A337267 and A337268. See also A337269.

Extensions

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