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A213954 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 3, n >= 2. 8
3, 4, 8, 6, 6, 8, 17, 14, 12, 10, 36, 32, 25, 18, 20, 12, 77, 68, 51, 36, 38, 20, 164, 142, 106, 72, 72, 38, 64, 28, 347, 298, 225, 146, 142, 74, 109, 46, 732, 628, 476, 302, 294, 148, 197, 82, 168, 64, 1543, 1324, 1003, 632, 614, 304, 385, 156, 277, 100, 3252, 2790, 2112, 1328, 1284, 634, 777, 312, 504, 174, 414, 136 (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 2 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....9...10...11...12
..n
..2.......3....4
..3.......8....6....6....8
..4......17...14...12...10
..5......36...32...25...18...20...12
..6......77...68...51...36...38...20
..7.....164..142..106...72...72...38...64...28
..8.....347..298..225..146..142...74..109...46
..9.....732..628..476..302..294..148..197...82..168...64
.10....1543.1324.1003..632..614..304..385..156..277..100
.11....3252.2790.2112.1328.1284..634..777..312..504..174..414..136
where k indicates the position of the start node in the quarter-rectangle.
For each n, the maximum value of k is 2*floor((n+1)/2).
Reading this array by rows gives the sequence.
LINKS
FORMULA
It appears that:
T(n,1) - 2*T(n-1,1) - T(n-4,1) - 2 = 0, n >= 6
T(n,2) - 2*T(n-1,2) - T(n-4,1) = 0, n >= 6
T(n,3) - 2*T(n-1,3) - T(n-4,1) = 0, n >= 10
T(n,4) - 2*T(n-1,4) - T(n-4,1) + 8 = 0, n >= 7
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 4 5
NT 3 4 3
3 4 3
To limit duplication, only the top left-hand corner 3 and the 4 to its right are stored in the sequence, i.e. T(2,1) = 3 and T(2,2) = 4.
CROSSREFS
Sequence in context: A253080 A050417 A327613 * A357067 A074212 A125715
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved

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Last modified April 17 04:22 EDT 2024. Contains 371756 sequences. (Running on oeis4.)