A335359 a(n) is the Y-coordinate of the n-th point of the Koch curve; sequence A335358 gives X-coordinates.

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

Offset: 0

Author

Rémy Sigrist, Jun 03 2020

Keywords

• nonn
• base

Comments

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

Y

/

/

0 ---- X

Examples

			The Koch curve starts (on a hexagonal lattice) as follows:
   .   .   .   .   .   .   +   .   .   .   .   .   .
                          /8\
     .   .   .   .   +---+   +---+   .   .   .   .
                     6\  7   9  /10
   .   .   .   +   .   +   .   +   .   +   .   .   .
              /2\     /5        \     / \
     .   +---+   +---+   .   .   +---+   +---+   .
         0   1   3   4           12  13  15  16
Hence, a(6) = a(7) = a(9) = a(10) = 2.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Wikipedia, Koch snowflake
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

Cf. A065359, A332206, A335358.

Programs

PARI

{ hex = [1,I,I-1,-1,-I,1-I]; z=0; for (n=0, 84, print1 (imag(z)", "); q=digits(n, 4); d=sum(k=1, #q, if (q[k]==1, +1, q[k]==2, -1, 0)); z+=hex[1+d%#hex]) }

Formula

a(n) = 0 iff n belongs to A332206.