A238009 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 3 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2), read by rows.

1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 3, 8, 3, 1, 3, 12, 8, 1, 4, 18, 22, 6, 1, 4, 24, 40, 22, 1, 5, 32, 73, 66, 10, 1, 5, 40, 112, 146, 48, 1, 6, 50, 172, 292, 174, 20, 1, 6, 60, 240, 516, 448, 116, 1, 7, 72, 335, 860, 1020, 464, 36, 1, 7, 84, 440, 1340, 2016, 1360, 256

Offset: 2

Author

Christopher Hunt Gribble, Feb 16 2014

Keywords

• tabf
• nonn

Examples

			The first 19 rows of T(n,k) are:
   n\k 0  1   2    3    4     5     6     7     8    9  10
   2   1  1
   3   1  1
   4   1  2   2
   5   1  2   4
   6   1  3   8    3
   7   1  3  12    8
   8   1  4  18   22    6
   9   1  4  24   40   22
  10   1  5  32   73   66    10
  11   1  5  40  112  146    48
  12   1  6  50  172  292   174    20
  13   1  6  60  240  516   448   116
  14   1  7  72  335  860  1020   464    36
  15   1  7  84  440 1340  2016  1360   256
  16   1  8  98  578 2010  3716  3400  1168    72
  17   1  8 112  728 2890  6336  7432  3840   584
  18   1  9 128  917 4046 10326 14864 10600  2920  136
  19   1  9 144 1120 5502 16016 27536 25344 10600 1280
  20   1 10 162 1368 7336 24066 48188 54992 31800 7080 272
		

Links

• Christopher Hunt Gribble and Andrew Howroyd, Rows n=2..60 of T(n,k) flattened (Rows n=2..20 from Christopher Hunt Gribble)
• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A034851, A226048, A102541, A226290, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Programs

PARI

T(n,k)={(2^k*binomial(n-1*k,k) + ((k%2==0)+(n%2==0||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-1*k-(k%2)-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/2), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Corrected C++ program and xrefs added by Christopher Hunt Gribble, Apr 25 2015