A213274 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 2, n >= 2.

4, 4, 4, 2, 4, 4, 6, 6, 4, 4, 6, 10, 10, 2, 4, 4, 6, 10, 14, 16, 8, 4, 4, 6, 10, 14, 20, 26, 18, 2, 4, 4, 6, 10, 14, 20, 30, 40, 34, 10, 4, 4, 6, 10, 14, 20, 30, 44, 60, 60, 28, 2, 4, 4, 6, 10, 14, 20, 30, 44, 64, 90, 100, 62, 12, 4, 4, 6, 10, 14, 20, 30, 44, 64, 94, 134, 160, 122, 40, 2

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

..n

..2....4

..3....4...4...2

..4....4...4...6...6

..5....4...4...6..10..10...2

..6....4...4...6..10..14..16...8

..7....4...4...6..10..14..20..26..18...2

..8....4...4...6..10..14..20..30..40..34..10

..9....4...4...6..10..14..20..30..44..60..60..28...2

.10....4...4...6..10..14..20..30..44..64..90.100..62..12

.11....4...4...6..10..14..20..30..44..64..94.134.160.122..40...2

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 + n mod 2)/2 for n >= 2.

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 a rectangle.

Links

Table of n, a(n) for n=2..81.

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.

Formula

The asymptotic sequence for the number of paths of each nodal length k for n >> 0 appears to be 2*A097333(1:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 3.

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 2 node rectangle.

Crossrefs

Cf. A213106, A213249.

Sequence in context: A220668 A109610 A067395 * A182565 A016708 A105724

Adjacent sequences: A213271 A213272 A213273 * A213275 A213276 A213277

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jun 08 2012

Status

approved