A213478 - OEIS

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

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OFFSET

2,1

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.

LINKS

Table of n, a(n) for n=2..72.

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.

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.

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