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.
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, 55, 19, 28, 94, 218, 294, 294, 218, 94, 28, 44, 140, 363, 506, 608, 506, 363, 140, 44, 60, 208, 536, 832, 1038, 1038, 832, 536, 208, 60, 85, 285, 785, 1240, 1695
Offset: 1
Examples
T(n,k) for 1<=n<=9 and 1<=k<=9 is: k 1 2 3 4 5 6 7 8 9 ... n 1 0 0 1 2 6 10 19 28 44 2 0 1 6 14 32 55 94 140 208 3 1 6 27 60 129 218 363 536 785 4 2 14 60 140 294 506 832 1240 1802 5 6 32 129 294 608 1038 1695 2516 3642 6 10 55 218 506 1038 1785 2902 4324 6242 7 19 94 363 832 1695 2902 4703 6992 10075 8 28 140 536 1240 2516 4324 6992 10416 14988 9 44 208 785 1802 3642 6242 10075 14988 21544
Links
- Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870
Crossrefs
Programs
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Maple
b := proc (n::integer, k::integer)::integer; (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); end proc; f := seq(seq(b(n, k - n + 1), n = 1 .. k), k = 1 .. 140);
Formula
Empirically,
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;
T(1,k) = A005993(k-3) = (k-1)*(2*(k-2)*k + 3*(1-(-1)^k))/24;
T(2,k) = A225972(k) = (k-1)*(2*k*(2*k-1) + 3*(1-(-1)^k))/12;
Extensions
Terms corrected and extended by Christopher Hunt Gribble, Apr 01 2015