A214399 Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.

6, 12, 14, 23, 24, 40, 42, 40, 68, 70, 70, 113, 116, 116, 122, 186, 190, 192, 202, 304, 310, 314, 334, 334, 495, 504, 512, 546, 552, 804, 818, 832, 890, 902, 912, 1304, 1326, 1350, 1446, 1470, 1490, 2113, 2148, 2188, 2346, 2388, 2428, 2434, 3422, 3478, 3544, 3802, 3874, 3944, 3966

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 1 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

..n

..2......6

..3.....12...14

..4.....23...24

..5.....40...42...40

..6.....68...70...70

..7....113..116..116..122

..8....186..190..192..202

..9....304..310..314..334..334

.10....495..504..512..546..552

.11....804..818..832..890..902..912

.12...1304.1326.1350.1446.1470.1490

.13...2113.2148.2188.2346.2388.2428.2434

.14...3422.3478.3544.3802.3874.3944.3966

.15...5540.5630.5738.6158.6278.6398.6442.6462

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

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

Reading this array by rows gives the sequence.

Links

Table of n, a(n) for n=2..56.

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 (N) occurs in a complete non-self-adjacent simple path is
N  0 1
   2 3
NT 6 6
   6 6
To limit duplication, only the top left-hand corner 6 is stored in the sequence, i.e. T(2,1) = 6.

Crossrefs

Cf. A213106, A213249, A213274, A213478, A214119, A214397.

Sequence in context: A315612 A315613 A246590 * A079946 A315614 A118586

Adjacent sequences: A214396 A214397 A214398 * A214400 A214401 A214402

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 15 2012

Extensions

Corrected by Christopher Hunt Gribble, Jul 19 2012

Status

approved