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A248059
Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing four 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.
6
0, 0, 0, 0, 1, 0, 1, 6, 6, 1, 3, 22, 39, 22, 3, 9, 60, 139, 139, 60, 9, 19, 135, 371, 476, 371, 135, 19, 38, 266, 813, 1253, 1253, 813, 266, 38, 66, 476, 1574, 2706, 3254, 2706, 1574, 476, 66, 110, 792, 2770, 5199, 6969, 6969, 5199, 2770, 792, 110, 170, 1245
OFFSET
1,8
LINKS
Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870
FORMULA
Empirically,
T(n,k) = (4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384;
T(1,k) = sum(A005993(i-4),i=1,k)
= sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);
T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.
EXAMPLE
T(n,k) for 1<=n<=9 and 1<=k<=9 is:
k 1 2 3 4 5 6 7 8 9 ...
n
1 0 0 0 1 3 9 19 38 66
2 0 1 6 22 60 135 266 476 792
3 0 6 39 139 371 813 1574 2770 4554
4 1 22 139 476 1253 2706 5199 9080 14857
5 3 60 371 1253 3254 6969 13294 23102 37637
6 9 135 813 2706 6969 14841 28197 48852 79401
7 19 266 1574 5199 13294 28197 53381 92266 149645
8 38 476 2770 9080 23102 48852 92266 159216 257878
9 66 792 4554 14857 37637 79401 149645 257878 417156
MAPLE
b := proc (n::integer, k::integer)::integer;
(4*k^4*n^4 - 24*k^3*n^3 + 2*k^4 + 12*k^3*n + 80*k^2*n^2 + 12*k*n^3 + 2*n^4 - 24*k^3 - 24*k^2*n - 24*k*n^2 - 24*n^3 + 40*k^2 - 102*k*n + 40*n^2 + 9 + (- 2*k^4 - 12*k^3*n + 24*k^3 + 24*k^2*n - 40*k^2 + 6*k*n - 9)*(-1)^n + (- 12*k*n^3 - 2*n^4 + 24*k*n^2 + 24*n^3 + 6*k*n - 40*n^2 - 9)*(-1)^k + (- 6*k*n + 9)*(-1)^k*(-1)^n)/384
end proc;
seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);
KEYWORD
tabl,nonn
AUTHOR
EXTENSIONS
Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015
STATUS
approved