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A213106
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Triangle T(n,k) of numbers/2 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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41
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4, 10, 32, 20, 82, 276, 36, 198, 898, 4028, 62, 456, 2770, 16840, 93664, 104, 1014, 8098, 65998, 483974, 3248120, 172, 2210, 22886, 250152, 2430726, 21169866, 177690360, 282, 4758, 63366, 931076, 12062348, 136925026, 1482885382, 15972807764
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OFFSET
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2,1
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COMMENTS
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The first 6 rows of the triangle are:
....k....2.....3.....4......5.......6........7
.n
.2.......4
.3......10....32
.4......20....82...276
.5......36...198...898...4028
.6......62...456..2770..16840...93664
.7.....104..1014..8098..65998..483974..3248120
Reading this triangle by rows gives the first 21 terms of the sequence.
One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of a rectangle.
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LINKS
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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) - 3*T(n-1, 3) + 2*T(n-2, 3) - T(n-4, 3) + T(n-5, 3) - 8*(n-4) = 0, n >= 9
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EXAMPLE
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T(2,2) = One half of the number of complete non-self-adjacent simple paths within a square lattice bounded by a 2 X 2 node rectangle.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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