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A213249
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Triangle T(n,k) of numbers of distinct shapes under rotation of non-extendable (complete) non-self-adjacent simple paths within a square lattice bounded by rectangles with nodal dimensions n and k, n >= k >= 2.
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40
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2, 8, 16, 18, 64, 134, 34, 170, 706, 1854, 60, 398, 2346, 13198, 41478, 102, 880, 6832, 55454, 382116, 1424988
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OFFSET
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2,1
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COMMENTS
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The triangle of numbers is:
....k....2....3.....4......5.......6........7
.n
.2.......2
.3.......8...16
.4......18...64...134
.5......34..170...706...1854
.6......60..398..2346..13198...41478
.7.....102..880..6832..55454..382116..1424988
The sequence is formed by reading the triangle by rows.
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LINKS
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Table of n, a(n) for n=2..22.
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.
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FORMULA
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Let T(n,k) denote an element of the triangle then the following recurrence relations appear to hold:
T(n, 2) - T(n-1, 2) - 2*A000045(n+1) = 0, n >= 3,
T(n, 3) - 2*T(n-1, 3) - T(n-4, 3) - 4*(n+11) = 0, n >= 7.
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EXAMPLE
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T(2,2) = The number of rotationally distinct complete non-self-adjacent simple path shapes within a 2 X 2 node rectangle.
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CROSSREFS
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Cf. A213106.
Sequence in context: A174882 A080095 A193219 * A155853 A256552 A031061
Adjacent sequences: A213246 A213247 A213248 * A213250 A213251 A213252
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KEYWORD
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nonn,tabl
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AUTHOR
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Christopher Hunt Gribble, Jun 07 2012
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STATUS
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approved
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