

A213274


Irregular array T(n,k) of numbers/2 of nonextendable (complete) nonselfadjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.


10



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
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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 nonselfadjacent paths in a square lattice bounded by various sizes of rectangle.
C. H. Gribble, Computes characteristics of complete nonselfadjacent 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..k2, C(k2j, floor(j/2)))), for k >= 3.


EXAMPLE

T(2,3) = One half of the number of complete nonselfadjacent 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



