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Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.
7

%I #9 Jul 19 2012 14:01:30

%S 6,12,14,23,24,40,42,40,68,70,70,113,116,116,122,186,190,192,202,304,

%T 310,314,334,334,495,504,512,546,552,804,818,832,890,902,912,1304,

%U 1326,1350,1446,1470,1490,2113,2148,2188,2346,2388,2428,2434,3422,3478,3544,3802,3874,3944,3966

%N Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths incorporating each of a minimal subset of nodes within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.

%C The subset of nodes is contained in the top left-hand quarter of the rectangle and has nodal dimensions floor((n+1)/2) and 1 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

%C ..n

%C ..2......6

%C ..3.....12...14

%C ..4.....23...24

%C ..5.....40...42...40

%C ..6.....68...70...70

%C ..7....113..116..116..122

%C ..8....186..190..192..202

%C ..9....304..310..314..334..334

%C .10....495..504..512..546..552

%C .11....804..818..832..890..902..912

%C .12...1304.1326.1350.1446.1470.1490

%C .13...2113.2148.2188.2346.2388.2428.2434

%C .14...3422.3478.3544.3802.3874.3944.3966

%C .15...5540.5630.5738.6158.6278.6398.6442.6462

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

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

%C Reading this array by rows gives the sequence.

%H C. H. Gribble, <a href="https://oeis.org/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="https://oeis.org/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>

%e When n = 2, the number of times (NT) each node in the rectangle (N) occurs in a complete non-self-adjacent simple path is

%e N 0 1

%e 2 3

%e NT 6 6

%e 6 6

%e To limit duplication, only the top left-hand corner 6 is stored in the sequence, i.e. T(2,1) = 6.

%Y Cf. A213106, A213249, A213274, A213478, A214119, A214397.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jul 15 2012

%E Corrected by _Christopher Hunt Gribble_, Jul 19 2012