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A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2. 5
2, 2, 4, 2, 2, 4, 6, 6, 2, 4, 6, 10, 10, 2, 2, 4, 6, 10, 14, 16, 8, 2, 4, 6, 10, 14, 20, 26, 18, 2, 2, 4, 6, 10, 14, 20, 30, 40, 34, 10, 2, 4, 6, 10, 14, 20, 30, 44, 60, 60, 28, 2, 2, 4, 6, 10, 14, 20, 30, 44, 64, 90, 100, 62, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

The irregular array of numbers is:

....k..3...4...5...6...7...8...9..10..11..12..13..14..15

..n

..2....2

..3....2...4...2

..4....2...4...6...6

..5....2...4...6..10..10...2

..6....2...4...6..10..14..16...8

..7....2...4...6..10..14..20..26..18...2

..8....2...4...6..10..14..20..30..40..34..10

..9....2...4...6..10..14..20..30..44..60..60..28...2

.10....2...4...6..10..14..20..30..44..64..90.100..62..12

where k is the path length in nodes. In an attempt to define the irregularity of the array, it appears that the maximum value of k is n + floor((n+1)/2) for n >= 2. Reading this array by rows gives the sequence.

LINKS

Table of n, a(n) for n=2..66.

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

The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k for n >> 0 appears to be 2*A097333(2:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 4.

EXAMPLE

T(2,3) = The number of distinct shapes under rotation of the complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 2 node rectangle.

CROSSREFS

Cf. A213106, A213249.

Sequence in context: A118232 A115070 A064145 * A331985 A261891 A321320

Adjacent sequences:  A213428 A213429 A213430 * A213432 A213433 A213434

KEYWORD

nonn,tabf

AUTHOR

Christopher Hunt Gribble, Jun 11 2012

STATUS

approved

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Last modified April 10 20:17 EDT 2021. Contains 342856 sequences. (Running on oeis4.)