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

1,3

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

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 Unpublished, 1976.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.

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

Example

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

Crossrefs

Cf. A003229, A203181.

Sequence in context: A358443 A175941 A317536 * A102477 A232582 A018074

Adjacent sequences: A203172 A203173 A203174 * A203176 A203177 A203178

Keyword

nonn

Author

R. H. Hardin, Dec 30 2011

Status

approved