A213342 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.

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, 90, 66, 184, 72, 236, 26, 4, 8, 20, 40, 78, 116, 192, 180, 354, 278, 530, 268, 546, 124, 32, 4, 8, 20, 40, 80, 122, 244, 336, 628, 628, 1130, 788, 1362, 878, 1168, 354, 292, 16

Offset: 2

Author

Christopher Hunt Gribble, Jun 09 2012

Keywords

• nonn
• tabf

Comments

The irregular array of numbers is:

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

.n

.2....4....4....6....6

.3....4....8...16...18...14....8...14

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

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

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

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

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 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.

Examples

			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.
		

Links

• C. H. Gribble, Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle.
• C. H. Gribble, Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively.

Crossrefs

Cf. A213106, A213249, A213274, A213089.