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A335703
Number of regions after n generations of symmetrized Conant Gasket.
11
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
0,2
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)
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)
PROG
(C++) See Links section.
CROSSREFS
For bisections see A337267 and A337268. See also A337269.
Sequence in context: A182558 A298204 A265581 * A107250 A050168 A331966
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(34)-a(35) from N. J. A. Sloane, Sep 06 2020 using Rémy Sigrist's C++ program.
STATUS
approved