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A248059 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. 6
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, 371, 135, 19, 38, 266, 813, 1253, 1253, 813, 266, 38, 66, 476, 1574, 2706, 3254, 2706, 1574, 476, 66, 110, 792, 2770, 5199, 6969, 6969, 5199, 2770, 792, 110, 170, 1245 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

LINKS

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

FORMULA

Empirically,

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;

T(1,k) = sum(A005993(i-4),i=1,k)

       = sum((i-2)*(2*(i-3)*(i-1) + 3*(1-(-1)^(i-1)))/24, i=1,k);

T(2,k) = A071239(k-1) = (k-1)*k*((k-1)^2+2)/6.

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      0      1      3      9     19     38      66

2       0      1      6     22     60    135    266    476     792

3       0      6     39    139    371    813   1574   2770    4554

4       1     22    139    476   1253   2706   5199   9080   14857

5       3     60    371   1253   3254   6969  13294  23102   37637

6       9    135    813   2706   6969  14841  28197  48852   79401

7      19    266   1574   5199  13294  28197  53381  92266  149645

8      38    476   2770   9080  23102  48852  92266 159216  257878

9      66    792   4554  14857  37637  79401 149645 257878  417156

MAPLE

b := proc (n::integer, k::integer)::integer;

(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

end proc;

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, A248011, A248016, A248060, A248017, A248027.

Sequence in context: A283100 A065493 A133890 * A111719 A228918 A200281

Adjacent sequences:  A248056 A248057 A248058 * A248060 A248061 A248062

KEYWORD

tabl,nonn

AUTHOR

Christopher Hunt Gribble, Sep 30 2014

EXTENSIONS

Terms corrected and extended by Christopher Hunt Gribble, Apr 06 2015

STATUS

approved

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Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)