A213478 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 2, n >= 2.

2, 3, 4, 5, 5, 8, 7, 6, 13, 10, 8, 21, 15, 11, 10, 34, 23, 16, 13, 55, 36, 24, 18, 16, 89, 57, 37, 26, 21, 144, 91, 58, 39, 29, 26, 233, 146, 92, 60, 42, 34, 377, 235, 147, 94, 63, 47, 42, 610, 379, 236, 149, 97, 68, 55, 987, 612, 380, 238, 152, 102, 76, 68, 1597, 989, 613, 382, 241, 157, 110, 89

Offset: 2

Author

Christopher Hunt Gribble, Jun 12 2012

Keywords

• nonn
• tabf

Comments

The subset of nodes approximately defines 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

..n

..2.......2

..3.......3...4

..4.......5...5

..5.......8...7...6

..6......13..10...8

..7......21..15..11..10

..8......34..23..16..13

..9......55..36..24..18..16

.10......89..57..37..26..21

.11.....144..91..58..39..29..26

.12.....233.146..92..60..42..34

.13.....377.235.147..94..63..47..42

.14.....610.379.236.149..97..68..55

.15.....987.612.380.238.152.102..76..68

.16....1597.989.613.382.241.157.110..89

where k indicates the position of the start 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.

Examples

			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
NT 2 2
   2 2
To limit duplication, only the top left-hand corner 2 is stored in the sequence, i.e. T(2,1) = 2.
		

Links

• 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.

Crossrefs

Cf. A213106, A213249, A213274.

Formula

Let T(n,k) denote an element of the irregular array then it appears that

T(n,k) = A000045(n-k+2), k = 0

T(n,k) = A000045(n-k+2) + A000045(k+1), k > 0.

Extensions

Improved Comments