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%I #27 Nov 30 2016 22:13:03
%S 0,0,0,1,1,1,2,6,6,2,6,14,27,14,6,10,32,60,60,32,10,19,55,129,140,129,
%T 55,19,28,94,218,294,294,218,94,28,44,140,363,506,608,506,363,140,44,
%U 60,208,536,832,1038,1038,832,536,208,60,85,285,785,1240,1695
%N Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing three 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.
%H Christopher Hunt Gribble, <a href="/A248011/b248011.txt">Table of n, a(n) for n = 1..9870</a>
%F Empirically,
%F T(n,k) = (4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)/96;
%F T(1,k) = A005993(k-3) = (k-1)*(2*(k-2)*k + 3*(1-(-1)^k))/24;
%F T(2,k) = A225972(k) = (k-1)*(2*k*(2*k-1) + 3*(1-(-1)^k))/12;
%F T(2,k) - T(1,k) = A199771(k-1) and A212561(k) = (k-1)*(6*k^2 + 3*(1-(-1)^k))/24.
%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 1 2 6 10 19 28 44
%e 2 0 1 6 14 32 55 94 140 208
%e 3 1 6 27 60 129 218 363 536 785
%e 4 2 14 60 140 294 506 832 1240 1802
%e 5 6 32 129 294 608 1038 1695 2516 3642
%e 6 10 55 218 506 1038 1785 2902 4324 6242
%e 7 19 94 363 832 1695 2902 4703 6992 10075
%e 8 28 140 536 1240 2516 4324 6992 10416 14988
%e 9 44 208 785 1802 3642 6242 10075 14988 21544
%p b := proc (n::integer, k::integer)::integer;
%p (4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)*(1/96);
%p end proc;
%p f := 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, A248016, A248059, A248060, A248017, A248027.
%K tabl,nonn
%O 1,7
%A _Christopher Hunt Gribble_, Sep 29 2014
%E Terms corrected and extended by _Christopher Hunt Gribble_, Apr 01 2015
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