

A213478


Irregular array T(n,k) of the numbers of nonextendable (complete) nonselfadjacent 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.


9



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 lefthand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 1 to capture all geometrically distinct counts.
The quarterrectangle 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 quarterrectangle. 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 nonselfadjacent paths in a square lattice bounded by various sizes of rectangle.
C. H. Gribble, Computes characteristics of complete nonselfadjacent 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(nk+2), k = 0
T(n,k) = A000045(nk+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 nonselfadjacent simple path is
SN 0 1
2 3
NT 2 2
2 2
To limit duplication, only the top lefthand corner 2 is stored in the sequence, i.e. T(2,1) = 2.


CROSSREFS

Cf. A213106, A213249, A213274.
Sequence in context: A017844 A303655 A011156 * A266449 A265536 A037849
Adjacent sequences: A213475 A213476 A213477 * A213479 A213480 A213481


KEYWORD

nonn,tabf


AUTHOR

Christopher Hunt Gribble, Jun 12 2012


EXTENSIONS

Improved Comments


STATUS

approved



