A203175 Number of nX2 0..2 arrays with every 1 immediately preceded by 0 to the left or above, no 0 immediately preceded by a 0, and every 2 immediately preceded by 0 1 to the left or above.

1, 1, 2, 4, 6, 10, 18, 30, 50, 86, 146, 246, 418, 710, 1202, 2038, 3458, 5862, 9938, 16854, 28578, 48454, 82162, 139318, 236226, 400550, 679186, 1151638, 1952738, 3311110, 5614386, 9519862, 16142082, 27370854, 46410578, 78694742, 133436450, 226257606

Offset: 1

Author

R. H. Hardin, Dec 30 2011

Keywords

• nonn

Comments

Column 2 of A203181.

It seems that for n>=1 a(n) equals the number of (n-1)-length binary words avoiding runs of zeros of length 1 (mod 3). - Milan Janjic, Feb 28 2015

Examples

			All solutions for n=5:
..0..1....0..1....0..1....0..1....0..1....0..1
..1..0....1..0....1..0....1..0....1..0....1..0
..0..1....0..1....0..1....2..1....2..1....0..1
..1..0....1..2....1..0....0..1....0..2....1..2
..2..1....2..0....0..1....1..0....1..0....0..1
		

References

• D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links in A003229 for an earlier version. See beta_n for this sequence. - N. J. A. Sloane, Jul 08 2014

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210
• Matthias Beck and Neville Robbins, Variations on a Generatingfunctional Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence, arXiv:1403.0665 [math.NT], 2014.
• Helena Verrill, On the Boundary of the Harter-Heighway dragon curve, arXiv:2407.17326 [math.CO], 2024.

Crossrefs

Cf. A003229, A203181.

Formula

Empirical: a(n) = a(n-1) + 2*a(n-3) = A003229(n-4)+A003229(n-2).

Empirical G.f.: -x*(1+x^2) / ( -1+x+2*x^3 ). - R. J. Mathar, Jul 02 2013