A213342 - OEIS

%I #18 Jun 19 2012 14:59:06

%S 4,4,6,6,4,8,16,18,14,8,14,4,8,20,36,44,24,40,16,84,4,8,20,40,72,80,

%T 90,66,184,72,236,26,4,8,20,40,78,116,192,180,354,278,530,268,546,124,

%U 32,4,8,20,40,80,122,244,336,628,628,1130,788,1362,878,1168,354,292,16

%N Irregular array T(n,k) of numbers/2 of non-extendable non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 4, 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....6....6

%C .3....4....8...16...18...14....8...14

%C .4....4....8...20...36...44...24...40...16...84

%C .5....4....8...20...40...72...80...90...66..184...72..236...26

%C .6....4....8...20...40...78..116..192..180..354..278..530..268..546..124...32

%C .7....4....8...20...40...80..122..244..336..628..628.1130..788.1362..878.1168..354..292...16

%C where k is the path length in nodes.

%C In an attempt to define the irregularity of the array, it appears that the maximum value of k is 3n for 2 <= n <= 3, 3n-1 for n = 4 and 3n - floor((n-2)/3) for n >= 5. 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 4 node rectangle.

%Y Cf. A213106, A213249, A213274, A213089.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jun 09 2012