

A213433


Irregular array T(n,k) of the numbers of distinct shapes under rotation of the nonextendable (complete) nonselfadjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 3, n >= 2.


4



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
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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..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 nonselfadjacent simple paths of each nodal length k, where n >= k1, 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.


LINKS

Table of n, a(n) for n=2..85.
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.


EXAMPLE

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


CROSSREFS

Cf. A213106, A213249, A213431.
Sequence in context: A210210 A117007 A340129 * A294354 A144049 A058384
Adjacent sequences: A213430 A213431 A213432 * A213434 A213435 A213436


KEYWORD

nonn,tabf


AUTHOR

Christopher Hunt Gribble, Jun 11 2012


EXTENSIONS

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STATUS

approved



