A248059 - OEIS

%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