A214121 - OEIS

%I #10 Jul 23 2012 12:45:37

%S 5,0,14,2,2,0,33,4,6,0,75,6,13,0,16,0,165,8,27,0,32,0,353,10,57,0,62,

%T 0,60,0,747,12,119,0,124,0,109,0,1577,14,247,0,250,0,206,0,184,0,3327,

%U 16,515,0,508,0,399,0,323,0,7015,18,1079,0,1046,0,790,0,590

%N Irregular array T(n,k) of the numbers of non-extendable (complete) non-self-adjacent simple paths ending 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 is contained in 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:

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

%C ..n

%C ..2.......5.....0

%C ..3......14.....2.....2.....0

%C ..4......33.....4.....6.....0

%C ..5......75.....6....13.....0....16.....0

%C ..6.....165.....8....27.....0....32.....0

%C ..7.....353....10....57.....0....62.....0....60.....0

%C ..8.....747....12...119.....0...124.....0...109.....0

%C ..9....1577....14...247.....0...250.....0...206.....0...184.....0

%C .10....3327....16...515.....0...508.....0...399.....0...323.....0

%C .11....7015....18..1079.....0..1046.....0...790.....0...590.....0...520.....0

%C .12...14785....20..2267.....0..2176.....0..1601.....0..1121.....0...877.....0

%C where k indicates the position of the end 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.

%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>

%F Let T(n,k) denote an element of the irregular array then it appears that

%F T(n,k) = 0, n >= 3, k = 2j, j >= 2,

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

%F T(n,2) = 2(n-2), n >= 2,

%F T(n,3) - 2T(n-1,3) - T(n-4,3) + 2(n-7) = 0, n >= 9,

%F T(n,5) - 2T(n-1,5) - T(n-4,5) + 8(n-7) = 0, n >= 10,

%F T(n,7) - 2T(n-1,7) - T(n-4,7) + 20(n-8) + 8 = 0, n >= 11,

%F T(n,9) - 2T(n-1,9) - T(n-4,9) + 46(n-9) + 30 = 0, n >= 13,

%F T(n,11) - 2T(n-1,11) - T(n-4,11) + 104(n-10) + 84 = 0, n >= 15,

%F T(n,13) - 2T(n-1,13) - T(n-4,13) + 226(n-11) + 202 = 0, n >= 15.

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

%e EN 0 1 2

%e    3 4 5

%e NT 5 0 5

%e    5 0 5

%e To limit duplication, only the top left-hand corner 5 and the 0 to its right are stored in the sequence, i.e. T(2,1) = 5 and T(2,2) = 0.

%Y Cf. A213106, A213249, A213089, A213954, A214119.

%K nonn,tabf

%O 2,1

%A _Christopher Hunt Gribble_, Jul 04 2012

%E Comment corrected by _Christopher Hunt Gribble_, Jul 22 2012