login
A213106
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.
41
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
OFFSET
2,1
COMMENTS
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.
FORMULA
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
EXAMPLE
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.
CROSSREFS
Sequence in context: A258041 A289447 A135831 * A360919 A015796 A318562
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved