A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the 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.

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

Offset: 2

Author

Christopher Hunt Gribble, Jun 11 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

..n

..2....2

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

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

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

..6....2...4...6..10..14..16...8

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

..8....2...4...6..10..14..20..30..40..34..10

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

.10....2...4...6..10..14..20..30..44..64..90.100..62..12

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 n + floor((n+1)/2) for n >= 2. Reading this array by rows gives the sequence.

Examples

			T(2,3) = The number of distinct shapes under rotation of the complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 2 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.

Formula

The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k for n >> 0 appears to be 2*A097333(2:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 4.