A213089 - OEIS

%I #31 Jun 22 2012 13:17:01

%S 4,4,2,4,8,12,0,8,4,8,16,18,14,8,14,4,8,16,22,42,24,42,22,18,4,8,16,

%T 22,48,60,82,90,66,34,24,2,4,8,16,22,50,66,132,160,218,120,122,56,36,

%U 4,4,8,16,22,52,68,144,222,334,406,302,288,198,88,52,6,4,8,16,22,54,70,152,238,416,574,810,642,760,456,320,136,72,8

%N Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.

%C The irregular array of numbers is:

%C ...k..3...4...5...6...7...8...9..10..11..12..13..14..15..16..17..18..19..20

%C .n

%C .2....4...4...2

%C .3....4...8..12...0...8

%C .4....4...8..16..18..14...8..14

%C .5....4...8..16..22..42..24..42..22..18

%C .6....4...8..16..22..48..60..82..90..66..34..24...2

%C .7....4...8..16..22..50..66.132.160.218.120.122..56..36...4

%C .8....4...8..16..22..52..68.144.222.334.406.302.288.198..88..52...6

%C .9....4...8..16..22..54..70.152.238.416..74.810.642.760.456.320.136..72...8

%C where k is the path length in nodes. In an attempt to define the irregularity of the array, it appears that the maximum value of k is 2n+1 for 2 <= n <= 6 and 2n+2 for n >= 7. Reading this array by rows gives the sequence. One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of the rectangle.

%H C. H. Gribble, <a href="https://oeis.org/wiki/Complete_non-self-adjacent_paths:Results_for_Square_Lattice">Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle.</a>

%H C. H. Gribble, <a href="https://oeis.org/wiki/Complete non-self-adjacent paths:Program">Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively.</a>

%e T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 3 node rectangle.

%Y Cf. A213106, A213249, A213274.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jun 08 2012