A214042 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 9, n >= 2.

55, 36, 24, 18, 16, 732, 476, 294, 197, 168, 628, 302, 148, 82, 64, 6115, 4840, 3979, 3349, 3076, 5170, 2597, 1718, 1595, 1564, 64904, 57210, 52820, 46787, 43294, 53478, 31544, 26459, 28472, 28700, 50228, 22432, 19802, 27924, 30696

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 5 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....13....14....15

.n

.2......55....36....24....18....16

.3.....732...476...294...197...168...628...302...148....82....64

.4....6115..4840..3979..3349..3076..5170..2597..1718..1595..1564

.5...64904.57210.52820.46787.43294.53478.31544.26459.28472.28700.50228.22432.19802.27924.30696

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

For each n, the maximum value of k is 5*floor((n+1)/2).

Reading this array by rows gives the sequence.

Links

Table of n, a(n) for n=2..41.

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 14 15 16 17
NT 55 36 24 18 16 18 24 36 55
   55 36 24 18 16 18 24 36 55
To limit duplication, only the top left-hand corner 34 and the 23, 16 and 13 to its right are stored in the sequence, i.e. T(2,1) = 34, T(2,2) = 23, T(2,3) = 16 and T(2,4) = 13.

Crossrefs

Cf. A213106, A213249, A213426, A213478, A213954, A214022, A214023, A214025, A214037, A214038

Sequence in context: A178509 A033375 A236416 * A112892 A232653 A291502

Adjacent sequences: A214039 A214040 A214041 * A214043 A214044 A214045

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 01 2012

Extensions

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

Status

approved