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.
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, 12, 16, 28, 48, 56, 56, 48, 28, 16, 20, 36, 65, 78, 88, 78, 65, 36, 20, 25, 45, 84, 106, 123, 123, 106, 84, 45, 25, 30, 55, 106, 136, 168, 171, 168, 136, 106, 55, 30
Offset: 1
Examples
T(n,k) for 1<=n<=11 and 1<=k<=11 is:
k 1 2 3 4 5 6 7 8 9 10 11 ...
.n
.1 0 1 2 4 6 9 12 16 20 25 30
.2 1 3 6 10 15 21 28 36 45 55 66
.3 2 6 13 22 34 48 65 84 106 130 157
.4 4 10 22 36 56 78 106 136 172 210 254
.5 6 15 34 56 88 123 168 216 274 335 406
.6 9 21 48 78 123 171 234 300 381 465 564
.7 12 28 65 106 168 234 321 412 524 640 777
.8 16 36 84 136 216 300 412 528 672 820 996
.9 20 45 106 172 274 381 524 672 856 1045 1270
10 25 55 130 210 335 465 640 820 1045 1275 1550
11 30 66 157 254 406 564 777 996 1270 1550 1885
Links
- Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870
Crossrefs
Formula
Empirically,
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.
T(1,k) = A002620(k) = floor(k^2/4).
T(2,k) = A000217(k) = k*(k+1)/2.
= T(1,k) + T(1,k+1) = floor(k^2/4) + floor((k+1)^2/4).
= k*(k+1) + floor((k-1)^2/4) - 1.
Extensions
Terms corrected and extended by Christopher Hunt Gribble, Apr 02 2015