A334483 a(n) is the X-coordinate of the n-th point of the Sierpinski arrowhead curve; sequence A334484 gives Y-coordinates.

0, 1, 1, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 6, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 3, 4, 5, 5, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 10, 9, 8, 8, 9, 9, 8, 8, 9, 10, 10, 11, 12, 13, 13, 12, 12, 13, 14, 14, 15

Offset: 0

Author

Rémy Sigrist, May 03 2020

Keywords

• nonn
• look

Comments

Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:

Y

/

/

0 ---- X

The Sierpinski arrowhead curve can be represented using an L-system.

Examples

			The Sierpinski arrowhead curve starts (on a hexagonal lattice) as follows:
                         +
                        27\
                           \
                            +
                           /26
                          /
                    +----+
                   /24   25
                  /
                 +          +----+
                23\        /20  19\
                   \      /        \
                    +----+          +
                    22   21        /18
                                  /
            +----+               +
           /4    5\             17\
          /        \               \
         +          +               +----+
         3\        /6               16  15\
           \      /                        \
            +    +          +----+          +
           /2    7\        /10  11\        /14
          /        \      /        \      /
    +----+          +----+          +----+
    0    1          8    9          12   13
- hence a(6) = a(7) = a(20) = a(21) = 2.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..2187
• Rémy Sigrist, Colored line plot of the first 3^7+1 points of the Sierpinski arrowhead curve (where the hue is function of the number of steps from the origin)
• Rémy Sigrist, PARI program for A334483
• Wikipedia, Arrowhead curve
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A262174, A307672, A334484.

Programs

PARI

See Links section.

Formula

A262174(1 + a(n) + A334484(n), 1 + A334484(n)) = 1 + n for any n >= 0.