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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. 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 (list; graph; refs; listen; history; text; internal format)
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

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.

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

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Last modified July 17 16:38 EDT 2019. Contains 325107 sequences. (Running on oeis4.)