cp's OEIS Frontend

A248011 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.

%I A248011 #27 Nov 30 2016 22:13:03
%S A248011 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 A248011 55,19,28,94,218,294,294,218,94,28,44,140,363,506,608,506,363,140,44,
%U A248011 60,208,536,832,1038,1038,832,536,208,60,85,285,785,1240,1695
%N A248011 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 A248011 Christopher Hunt Gribble, <a href="/A248011/b248011.txt">Table of n, a(n) for n = 1..9870</a>
%F A248011 Empirically,
%F A248011 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 A248011 T(1,k) = A005993(k-3) = (k-1)*(2*(k-2)*k + 3*(1-(-1)^k))/24;
%F A248011 T(2,k) = A225972(k) = (k-1)*(2*k*(2*k-1) + 3*(1-(-1)^k))/12;
%F A248011 T(2,k) - T(1,k) = A199771(k-1) and A212561(k) = (k-1)*(6*k^2 + 3*(1-(-1)^k))/24.
%e A248011 T(n,k) for 1<=n<=9 and 1<=k<=9 is:
%e A248011    k    1     2     3     4     5     6     7     8     9 ...
%e A248011 n
%e A248011 1       0     0     1     2     6    10    19    28    44
%e A248011 2       0     1     6    14    32    55    94   140   208
%e A248011 3       1     6    27    60   129   218   363   536   785
%e A248011 4       2    14    60   140   294   506   832  1240  1802
%e A248011 5       6    32   129   294   608  1038  1695  2516  3642
%e A248011 6      10    55   218   506  1038  1785  2902  4324  6242
%e A248011 7      19    94   363   832  1695  2902  4703  6992 10075
%e A248011 8      28   140   536  1240  2516  4324  6992 10416 14988
%e A248011 9      44   208   785  1802  3642  6242 10075 14988 21544
%p A248011 b := proc (n::integer, k::integer)::integer;
%p A248011 (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 A248011 end proc;
%p A248011 f := seq(seq(b(n, k - n + 1), n = 1 .. k), k = 1 .. 140);
%Y A248011 Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248016, A248059, A248060, A248017, A248027.
%K A248011 tabl,nonn
%O A248011 1,7
%A A248011 _Christopher Hunt Gribble_, Sep 29 2014
%E A248011 Terms corrected and extended by _Christopher Hunt Gribble_, Apr 01 2015