A171158 The number of walks from (0,0,0) to (n,n,n) with steps that increment one to three coordinates and having the property that no two consecutive steps are orthogonal.

1, 1, 19, 235, 3181, 44725, 648439, 9614329, 145020445, 2217212539, 34269961873, 534449721793, 8397498847645, 132785160326593, 2111135363144743, 33723822603109987, 540949658114010583, 8708952402795685879, 140665766088396528829, 2278642960112808284773

Offset: 0

Author

Lee A. Newberg, Dec 04 2009

Keywords

• nonn
• walk

Comments

a(n) is also the number of standard sequence alignments of three strings of length n, counting only those alignments with the property that, for every pair of consecutive alignment columns, there is at least one sequence that contributes a non-gap to both columns. That is, a(n) counts only those standard alignments with a column order that can be unambiguously reconstructed from the knowledge of all pairings, where a pairing is, e.g., that some i-th position of some string x is in the same column as some j-th position of some string y. - Lee A. Newberg, Dec 11 2009

Examples

			For n = 2, the 19 walks are:
000 -> 001 -> 012 -> 122 -> 222
000 -> 001 -> 102 -> 212 -> 222
000 -> 001 -> 112 -> 222
000 -> 010 -> 021 -> 122 -> 222
000 -> 010 -> 120 -> 221 -> 222
000 -> 010 -> 121 -> 222
000 -> 011 -> 112 -> 222
000 -> 011 -> 121 -> 222
000 -> 011 -> 122 -> 222
000 -> 100 -> 201 -> 212 -> 222
000 -> 100 -> 210 -> 221 -> 222
000 -> 100 -> 211 -> 222
000 -> 101 -> 112 -> 222
000 -> 101 -> 211 -> 222
000 -> 101 -> 212 -> 222
000 -> 110 -> 121 -> 222
000 -> 110 -> 211 -> 222
000 -> 110 -> 221 -> 222
000 -> 111 -> 222
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..150

Crossrefs

See A171155 for the number of such walks in two dimensions.

See A171563 for the number of such walks in four dimensions. - Lee A. Newberg, Dec 11 2009

Formula

a(n) ~ c * d^n / n, where d = 17.073685937995..., c = 0.171212682922... . - Vaclav Kotesovec, Sep 10 2014

Extensions

Extended beyond a(10) by Alois P. Heinz, Jan 22 2013