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%I #38 May 30 2017 12:14:22
%S 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,
%T 73,66,10,1,5,40,112,146,48,1,6,50,172,292,174,20,1,6,60,240,516,448,
%U 116,1,7,72,335,860,1020,464,36,1,7,84,440,1340,2016,1360,256
%N 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.
%H Christopher Hunt Gribble and Andrew Howroyd, <a href="/A238009/b238009.txt">Rows n=2..60 of T(n,k) flattened</a> (Rows n=2..20 from Christopher Hunt Gribble)
%H Christopher Hunt Gribble, <a href="/A238009/a238009_1.cpp.txt">C++ program</a>
%e The first 19 rows of T(n,k) are:
%e n\k 0 1 2 3 4 5 6 7 8 9 10
%e 2 1 1
%e 3 1 1
%e 4 1 2 2
%e 5 1 2 4
%e 6 1 3 8 3
%e 7 1 3 12 8
%e 8 1 4 18 22 6
%e 9 1 4 24 40 22
%e 10 1 5 32 73 66 10
%e 11 1 5 40 112 146 48
%e 12 1 6 50 172 292 174 20
%e 13 1 6 60 240 516 448 116
%e 14 1 7 72 335 860 1020 464 36
%e 15 1 7 84 440 1340 2016 1360 256
%e 16 1 8 98 578 2010 3716 3400 1168 72
%e 17 1 8 112 728 2890 6336 7432 3840 584
%e 18 1 9 128 917 4046 10326 14864 10600 2920 136
%e 19 1 9 144 1120 5502 16016 27536 25344 10600 1280
%e 20 1 10 162 1368 7336 24066 48188 54992 31800 7080 272
%o (C++) See Gribble link.
%o (PARI)
%o 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}
%o for(n=2,20,for(k=0,floor(n/2), print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 29 2017
%Y 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.
%K tabf,nonn
%O 2,6
%A _Christopher Hunt Gribble_, Feb 16 2014
%E Corrected C++ program and xrefs added by _Christopher Hunt Gribble_, Apr 25 2015
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