A265686 Number of shapes of grid-filling curves of order 6*n+1 (on the tri-hexagonal grid) with turns by +-60 and +-120 degrees that are generated by Lindenmayer-systems with just one symbol apart from the turns.

1, 3, 7, 10, 63, 157, 456, 1830, 0, 8538, 23114, 61804, 165123, 0, 2339000

Offset: 1

Author

Joerg Arndt, Dec 13 2015

Keywords

• nonn
• hard
• more

Comments

Such curves exist only for n such that 6*n+1 is a term of A003136, these values 6*n+1 are given in A260682.

The first such curve, the only one of order 7, was discovered by Jeffrey Ventrella, see page 105 in the Ventrella reference.

Links

• Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
• Jeffrey J. Ventrella, Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012).

Crossrefs

Cf. A234434 (shapes on the triangular grid), A265685 (shapes on the square grid).

Extensions

a(11) - a(15) from Joerg Arndt, Feb 10 2019