A229216 If 1, 2, and 3 represent the three 2D vectors (1,0), (0.5,sqrt(3)) and (-0.5,sqrt(3)) and -1, -2 and -3 are the negation of these vectors, then this sequence represents Koch's snowflake.

1, -3, 2, 1, -3, -2, 1, -3, 2, 1, 3, 2, 1, -3, 2, 1, -3, -2, 1, -3, -2, -1, -3, -2, 1, -3, 2, 1, -3, -2, 1, -3, 2, 1, 3, 2, 1, -3, 2, 1, 3, 2, -1, 3, 2, 1, 3, 2, 1, -3, 2, 1, -3, -2, 1, -3, 2, 1, 3, 2, 1, -3, 2, 1, 3, 2, -1, 3, 2, 1, 3, 2, -1, 3, -2, -1, 3, 2, -1, 3, 2, 1, 3, 2, 1, -3, 2, 1, 3, 2, -1, 3, 2, 1, 3, 2, -1, 3, -2, -1, 3, 2, -1, 3, -2, -1, -3, -2, -1

Offset: 1

Author

Arie Bos, Sep 25 2013

Keywords

• easy
• sign

Comments

The sequence is generated by:

P(1) = 1,-3,2,1,

P(2) = 2,1,3,2,

P(3) = 3,2,-1,3,

P(-1) = -1,3,-2,-1,

P(-2) = -2,-1,-3,-2,

P(-3) = -3,-2,1,-3 (we have P(-x)=-P(x)), and 1, 3, -2 is the start.

Examples

			Start 1,3,-2,
in the first step 1,-3,2,1,3,2,-1,3,-2,-1,-3,-2 and
in the second step 1, -3, 2, 1, -3, -2, 1, -3, 2, 1, 3, 2, ..., -2, -1, -3, -2.
With each step the length increases by a factor 4.
		

Links

• Arie Bos, Index notation of grid graphs, arXiv:1210.7123 [cs.CG], 2012.
• Skylyn Irby, Sandra Spiroff, On conditionally defined Fibonacci and Lucas sequences and periodicity, Bull. Korean Math. Soc. (2020) Vol. 57, No. 4, 1033-1048.
• Wikipedia, Koch snowflake

Crossrefs

Cf. A229217.