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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. 6
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, 794, 198, 12, 28, 508, 2196, 3912, 3912, 2196, 508, 28, 66, 1092, 5231, 10626, 13462, 10626, 5231, 1092, 66, 126, 2156, 10808, 24648, 35787, 35787, 24648, 10808, 2156, 126 (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^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

+ (- 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

+ (- 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

+ (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;

T(1,k) = A005995(k-5) = (k-3)*(k-1)*((k-4)*(k-2)*2*k + 15*(1-(-1)^k))/480;

T(2,k) = A222715(k) = (k-2)*(k-1)*((2*k-3)(2*k-1)*2*k + 15*(1-(-1)^k))/120.

EXAMPLE

T(n,k) for 1<=n<=8 and 1<=k<=8 is:

.  k   1      2      3      4      5      6      7       8 ...

n

1      0      0      0      0      1      3     12      28

2      0      0      2     14     66    198    508    1092

3      0      2     39    208    794   2196   5231   10808

4      0     14    208   1092   3912  10626  24648   50344

5      1     66    794   3912  13462  35787  81648  164980

6      3    198   2196  10626  35787  94248 212988  428076

7     12    508   5231  24648  81648 212988 477903  955856

8     28   1092  10808  50344 164980 428076 955856 1906128

MAPLE

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

(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

   + (- 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

   + (- 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

   + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;

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, A248059, A248060, A248027.

Sequence in context: A225678 A141720 A353449 * A244606 A273127 A103272

Adjacent sequences:  A248014 A248015 A248016 * A248018 A248019 A248020

KEYWORD

tabl,nonn

AUTHOR

Christopher Hunt Gribble, Sep 30 2014

EXTENSIONS

Terms corrected and extended by Christopher Hunt Gribble, Apr 16 2015

STATUS

approved

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Last modified September 29 12:57 EDT 2022. Contains 357090 sequences. (Running on oeis4.)