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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.
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%I #20 Oct 16 2018 11:24:01

%S 3,4,8,6,6,8,17,14,12,10,36,32,25,18,20,12,77,68,51,36,38,20,164,142,

%T 106,72,72,38,64,28,347,298,225,146,142,74,109,46,732,628,476,302,294,

%U 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

%N 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.

%C 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.

%C The quarter-rectangle is read by rows.

%C The irregular array of numbers is:

%C ....k.....1....2....3....4....5....6....7....8....9...10...11...12

%C ..n

%C ..2.......3....4

%C ..3.......8....6....6....8

%C ..4......17...14...12...10

%C ..5......36...32...25...18...20...12

%C ..6......77...68...51...36...38...20

%C ..7.....164..142..106...72...72...38...64...28

%C ..8.....347..298..225..146..142...74..109...46

%C ..9.....732..628..476..302..294..148..197...82..168...64

%C .10....1543.1324.1003..632..614..304..385..156..277..100

%C .11....3252.2790.2112.1328.1284..634..777..312..504..174..414..136

%C where k indicates the position of the start node in the quarter-rectangle.

%C For each n, the maximum value of k is 2*floor((n+1)/2).

%C Reading this array by rows gives the sequence.

%H C. H. Gribble, <a href="/wiki/Complete_non-self-adjacent_paths:Results_for_Square_Lattice">Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle.</a>

%H C. H. Gribble, <a href="/wiki/Complete non-self-adjacent paths:Program">Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively.</a>

%F It appears that:

%F T(n,1) - 2*T(n-1,1) - T(n-4,1) - 2 = 0, n >= 6

%F T(n,2) - 2*T(n-1,2) - T(n-4,1) = 0, n >= 6

%F T(n,3) - 2*T(n-1,3) - T(n-4,1) = 0, n >= 10

%F T(n,4) - 2*T(n-1,4) - T(n-4,1) + 8 = 0, n >= 7

%e 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

%e SN 0 1 2

%e 3 4 5

%e NT 3 4 3

%e 3 4 3

%e 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.

%Y Cf. A213106, A213249, A213089, A213478.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jun 30 2012