A213433 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 3, n >= 2.

2, 4, 2, 2, 4, 6, 0, 4, 2, 4, 10, 18, 8, 8, 14, 2, 4, 10, 22, 34, 22, 36, 22, 18, 2, 4, 10, 22, 38, 56, 68, 80, 58, 34, 24, 2, 2, 4, 10, 22, 38, 60, 110, 138, 188, 106, 108, 54, 36, 4, 2, 4, 10, 22, 38, 60, 114, 188, 280, 360, 248, 254, 174, 84, 52, 6, 2, 4, 10, 22, 38, 60, 114, 192, 338, 494, 694, 534, 642, 402, 282, 130, 72, 8

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..16..17..18..19..20

.n

.2....2...4...2

.3....2...4...6...0...4

.4....2...4..10..18...8...8..14

.5....2...4..10..22..34..22..36..22..18

.6....2...4..10..22..38..56..68..80..58..34..24...2

.7....2...4..10..22..38..60.110.138.188.106.108..54..36...4

.8....2...4..10..22..38..60.114.188.280.360.248.254.174..84..52...6

.9....2...4..10..22..38..60.114.192.338.494.694.534.642.402.282.130..72...8

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 2n+1 for 2 <= n <= 6 and 2n+2 for n >= 7. Reading this array by rows gives the sequence.

The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k, where n >= k-1, is 2, 4, 10, 22, 38, 60, 114, 192, 342, 564, 956, 1584, 2686, 4524, 7684, 12968 for which there appears to be no obvious formula.

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 3 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, A213431.

Extensions

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