A214037 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 7, n >= 2.

21, 15, 11, 10, 164, 106, 72, 64, 142, 72, 38, 28, 888, 695, 607, 602, 780, 385, 258, 270, 5600, 4795, 4453, 4412, 4829, 2792, 2285, 2556, 4650, 2036, 1712, 2248, 35971, 30709, 27591, 26574, 30070, 18037, 14507, 15318, 27638, 13744, 13851, 17846

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 4 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.......21....15....11....10

.3......164...106....72....64....142...72....38....28

.4......888...695...607...602...780...385...258...270

.5.....5600..4795..4453..4412..4829..2792..2285..2556..4650..2036..1712..2248

.6....35971.30709.27591.26574.30070.18037.14507.15318.27638.13744.13851.17846

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

For each n, the maximum value of k is 4*floor((n+1)/2).

Reading this array by rows gives the sequence.

Links

Table of n, a(n) for n=2..45.

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 10 11 12 13
NT 21 15 11 10 11 15 21
   21 15 11 10 11 15 21
To limit duplication, only the top left-hand corner 21 and the 15, 11 and 10 to its right are stored in the sequence, i.e. T(2,1) = 21, T(2,2) = 15, T(2,3) = 11 and T(2,4) = 10.

Crossrefs

Cf. A213106, A213249, A213379, A213478, A213954, A214022, A214023, A214025

Sequence in context: A380671 A068015 A040422 * A321281 A048933 A142590

Adjacent sequences: A214034 A214035 A214036 * A214038 A214039 A214040

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 01 2012

Status

approved