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

2,1

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.

Links

Table of n, a(n) for n=2..66.

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.

Example

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.

Crossrefs

Cf. A213106, A213249, A213274, A213089.

Sequence in context: A016710 A225134 A121064 * A019559 A365198 A274636

Adjacent sequences: A213339 A213340 A213341 * A213343 A213344 A213345

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jun 09 2012

Status

approved