A214121 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.

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, 0, 60, 0, 747, 12, 119, 0, 124, 0, 109, 0, 1577, 14, 247, 0, 250, 0, 206, 0, 184, 0, 3327, 16, 515, 0, 508, 0, 399, 0, 323, 0, 7015, 18, 1079, 0, 1046, 0, 790, 0, 590

Offset

2,1

Comments

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:

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

..n

..2.......5.....0

..3......14.....2.....2.....0

..4......33.....4.....6.....0

..5......75.....6....13.....0....16.....0

..6.....165.....8....27.....0....32.....0

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

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

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

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

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

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

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.

Links

Table of n, a(n) for n=2..68.

C. H. Gribble, Computed characteristics of complete non-self-adjacent paths in a square lattice bounded by various sizes of rectangle.

C. H. Gribble, Computes characteristics of complete non-self-adjacent paths in square and cubic lattices bounded by various sizes of rectangle and rectangular cuboid respectively.

Formula

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

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

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

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

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

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

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

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

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

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

Example

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
EN 0 1 2
   3 4 5
NT 5 0 5
   5 0 5
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.

Crossrefs

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

Sequence in context: A292105 A052401 A222946 * A024418 A167297 A290867

Adjacent sequences: A214118 A214119 A214120 * A214122 A214123 A214124

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Jul 04 2012

Extensions

Comment corrected by Christopher Hunt Gribble, Jul 22 2012

Status

approved