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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
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.
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
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
Improved Comments
STATUS
approved