A214023 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths starting at each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 5, n >= 2.

8, 7, 6, 36, 25, 20, 32, 18, 12, 122, 102, 94, 110, 52, 32, 436, 395, 394, 395, 220, 154, 394, 154, 80, 1580, 1414, 1402, 1381, 813, 596, 1365, 652, 432, 5600, 4829, 4650, 4795, 2792, 2036, 4453, 2285, 1712, 4412, 2556, 2248, 19287, 16131, 15246, 16735, 9444, 6758, 15113, 7697, 5858, 13878, 8612, 8496

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........8.....7.....6

..3.......36....25....20....32....18....12

..4......122...102....94...110....52....32

..5......436...395...394...395...220...154...394...154....80

..6.....1580..1414..1402..1381...813...596..1365...652...432

..7.....5600..4829..4650..4795..2792..2036..4453..2285..1712..4412..2556..2248

..8....19287.16131.15246.16735..9444..6758.15113..7697..5858.13878..8612..8496

where k indicates the position of the start 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..58.

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 start node (SN) of a complete non-self-adjacent simple path is
SN 0 1 2 3 4
   5 6 7 8 9
NT 8 7 6 7 8
   8 7 6 7 8
To limit duplication, only the top left-hand corner 8 and the 7 and 6 to its right are stored in the sequence, i.e. T(2,1) = 8, T(2,2) = 7 and T(2,3) = 6.

Crossrefs

Cf. A213106, A213249, A213375, A213478, A213954, A214022

Sequence in context: A247095 A124597 A347903 * A154209 A115373 A021118

Adjacent sequences: A214020 A214021 A214022 * A214024 A214025 A214026

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 01 2012

Status

approved