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A244306 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing two 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

%I A244306 #26 Nov 30 2016 22:18:17
%S A244306 0,1,1,2,3,2,4,6,6,4,6,10,13,10,6,9,15,22,22,15,9,12,21,34,36,34,21,
%T A244306 12,16,28,48,56,56,48,28,16,20,36,65,78,88,78,65,36,20,25,45,84,106,
%U A244306 123,123,106,84,45,25,30,55,106,136,168,171,168,136,106,55,30
%N A244306 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing two 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.
%H A244306 Christopher Hunt Gribble, <a href="/A244306/b244306.txt">Table of n, a(n) for n = 1..9870</a>
%F A244306 Empirically,
%F A244306 T(n,k) = (4*k^2*n^2 + 2*k^2 + 8*k*n + 2*n^2 - 4*k - 4*n - 1 - (2*k^2 - 4*k - 1)*(-1)^n - (2*n^2 - 4*n - 1)*(-1)^k - (-1)^k*(-1)^n)/32.
%F A244306 T(1,k) = A002620(k) = floor(k^2/4).
%F A244306 T(2,k) = A000217(k) = k*(k+1)/2.
%F A244306        = T(1,k) + T(1,k+1) = floor(k^2/4) + floor((k+1)^2/4).
%F A244306 T(3,k) = 2*A000217(k) + A024206(k-2)
%F A244306        = k*(k+1) + floor((k-1)^2/4) - 1.
%e A244306 T(n,k) for 1<=n<=11 and 1<=k<=11 is:
%e A244306     k  1    2    3    4    5    6    7    8    9   10   11 ...
%e A244306 .n
%e A244306 .1     0    1    2    4    6    9   12   16   20   25   30
%e A244306 .2     1    3    6   10   15   21   28   36   45   55   66
%e A244306 .3     2    6   13   22   34   48   65   84  106  130  157
%e A244306 .4     4   10   22   36   56   78  106  136  172  210  254
%e A244306 .5     6   15   34   56   88  123  168  216  274  335  406
%e A244306 .6     9   21   48   78  123  171  234  300  381  465  564
%e A244306 .7    12   28   65  106  168  234  321  412  524  640  777
%e A244306 .8    16   36   84  136  216  300  412  528  672  820  996
%e A244306 .9    20   45  106  172  274  381  524  672  856 1045 1270
%e A244306 10    25   55  130  210  335  465  640  820 1045 1275 1550
%e A244306 11    30   66  157  254  406  564  777  996 1270 1550 1885
%Y A244306 Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244307, A248011, A248016, A248059, A248060, A248017, A248027.
%K A244306 tabl,nonn
%O A244306 1,4
%A A244306 _Christopher Hunt Gribble_, Jun 25 2014
%E A244306 Terms corrected and extended by _Christopher Hunt Gribble_, Apr 02 2015