A213070 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths ending at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 6, n >= 2.

31, 0, 0, 165, 27, 32, 8, 0, 0, 720, 187, 236, 104, 30, 108, 3431, 992, 1179, 746, 251, 580, 920, 352, 1210, 16608, 4361, 5027, 4361, 1094, 2043, 5027, 2043, 6268, 76933, 17601, 20009, 21068, 3675, 7213, 26181, 9258, 26414, 25090, 10048, 32132

Offset

2,1

Comments

The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 3 to capture all geometrically distinct counts.

The quarter-rectangle is read by rows.

The irregular array of numbers is:

...k.....1.....2.....3.....4.....5.....6.....7.....8.....9....10....11....12

.n

.2......31.....0.....0

.3.....165....27....32.....8.....0.....0

.4.....720...187...236...104....30...108

.5....3431...992..1179...746...251...580...920...352..1210

.6...16608..4361..5027..4361..1094..2043..5027..2043..6268

.7...76933.17601.20009.21068..3675..7213.26181..9258.26414.25090.10048.32132

where k indicates the position of the end node in the quarter-rectangle.

For each n, the maximum value of k is 3*floor((n+1)/2).

Reading this array by rows gives the sequence.

Links

Table of n, a(n) for n=2..46.

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.

Example

When n = 2, the number of times (NT) each node in the rectangle is the end node (EN) of a complete non-self-adjacent simple path is
EN  0  1  2  3  4  5
    6  7  8  9 10 11
NT 31  0  0  0  0 31
   31  0  0  0  0 31
To limit duplication, only the top left-hand corner 31 and the two zeros to its right are stored in the sequence, i.e. T(2,1) = 31, T(2,2) = 0 and T(2,3) = 0.

Crossrefs

Cf. A213106, A213249, A213379, A214025, A214119, A214121, A214122, A214359.

Sequence in context: A028363 A091511 A324684 * A221432 A085012 A086019

Adjacent sequences: A213067 A213068 A213069 * A213071 A213072 A213073

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 13 2012

Extensions

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

Status

approved