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A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.

%I A248017 #31 Nov 30 2016 22:12:07
%S A248017 0,0,0,0,0,0,0,2,2,0,1,14,39,14,1,3,66,208,208,66,3,12,198,794,1092,
%T A248017 794,198,12,28,508,2196,3912,3912,2196,508,28,66,1092,5231,10626,
%U A248017 13462,10626,5231,1092,66,126,2156,10808,24648,35787,35787,24648,10808,2156,126
%N A248017 Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.
%H A248017 Christopher Hunt Gribble, <a href="/A248017/b248017.txt">Table of n, a(n) for n = 1..9870</a>
%F A248017 Empirically,
%F A248017 T(n,k) = (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4 - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3 - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2 + 48*k + 48*n + 45
%F A248017 + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3 + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n - 45)*(-1)^k
%F A248017 + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k - 45)*(-1)^n
%F A248017 + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
%F A248017 T(1,k) = A005995(k-5) = (k-3)*(k-1)*((k-4)*(k-2)*2*k + 15*(1-(-1)^k))/480;
%F A248017 T(2,k) = A222715(k) = (k-2)*(k-1)*((2*k-3)(2*k-1)*2*k + 15*(1-(-1)^k))/120.
%e A248017 T(n,k) for 1<=n<=8 and 1<=k<=8 is:
%e A248017 .  k   1      2      3      4      5      6      7       8 ...
%e A248017 n
%e A248017 1      0      0      0      0      1      3     12      28
%e A248017 2      0      0      2     14     66    198    508    1092
%e A248017 3      0      2     39    208    794   2196   5231   10808
%e A248017 4      0     14    208   1092   3912  10626  24648   50344
%e A248017 5      1     66    794   3912  13462  35787  81648  164980
%e A248017 6      3    198   2196  10626  35787  94248 212988  428076
%e A248017 7     12    508   5231  24648  81648 212988 477903  955856
%e A248017 8     28   1092  10808  50344 164980 428076 955856 1906128
%p A248017 b := proc (n::integer, k::integer)::integer;
%p A248017 (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n
%p A248017    + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4
%p A248017    - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3
%p A248017    - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2
%p A248017    + 48*k + 48*n + 45
%p A248017    + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3
%p A248017       + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n
%p A248017       - 45)*(-1)^k
%p A248017    + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n
%p A248017       - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n
%p A248017       - 48*k - 45)*(-1)^n
%p A248017    + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
%p A248017 end proc;
%p A248017 seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);
%Y A248017 Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248027.
%K A248017 tabl,nonn
%O A248017 1,8
%A A248017 _Christopher Hunt Gribble_, Sep 30 2014
%E A248017 Terms corrected and extended by _Christopher Hunt Gribble_, Apr 16 2015