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%I #27 Nov 30 2016 22:11:19
%S 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,
%T 371,135,19,38,266,813,1253,1253,813,266,38,66,476,1574,2706,3254,
%U 2706,1574,476,66,110,792,2770,5199,6969,6969,5199,2770,792,110,170,1245
%N 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.
%H Christopher Hunt Gribble, <a href="/A248059/b248059.txt">Table of n, a(n) for n = 1..9870</a>
%F Empirically,
%F 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;
%F T(1,k) = sum(A005993(i-4),i=1,k)
%F = sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);
%F T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.
%e T(n,k) for 1<=n<=9 and 1<=k<=9 is:
%e k 1 2 3 4 5 6 7 8 9 ...
%e n
%e 1 0 0 0 1 3 9 19 38 66
%e 2 0 1 6 22 60 135 266 476 792
%e 3 0 6 39 139 371 813 1574 2770 4554
%e 4 1 22 139 476 1253 2706 5199 9080 14857
%e 5 3 60 371 1253 3254 6969 13294 23102 37637
%e 6 9 135 813 2706 6969 14841 28197 48852 79401
%e 7 19 266 1574 5199 13294 28197 53381 92266 149645
%e 8 38 476 2770 9080 23102 48852 92266 159216 257878
%e 9 66 792 4554 14857 37637 79401 149645 257878 417156
%p b := proc (n::integer, k::integer)::integer;
%p (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
%p end proc;
%p seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);
%Y Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248060, A248017, A248027.
%K tabl,nonn
%O 1,8
%A _Christopher Hunt Gribble_, Sep 30 2014
%E Terms corrected and extended by _Christopher Hunt Gribble_, Apr 06 2015
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