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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. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

LINKS

Christopher Hunt Gribble, Table of n, a(n) for n = 1..9870

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;

T(2,k) - T(1,k) = A199771(k-1) and A212561(k) = (k-1)*(6*k^2 + 3*(1-(-1)^k))/24.

EXAMPLE

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

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);

CROSSREFS

Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248016, A248059, A248060, A248017, A248027.

Sequence in context: A175994 A283613 A141327 * A282729 A011386 A097412

Adjacent sequences:  A248008 A248009 A248010 * A248012 A248013 A248014

KEYWORD

tabl,nonn

AUTHOR

Christopher Hunt Gribble, Sep 29 2014

EXTENSIONS

Terms corrected and extended by Christopher Hunt Gribble, Apr 01 2015

STATUS

approved

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Last modified October 29 21:58 EDT 2020. Contains 338074 sequences. (Running on oeis4.)