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A238190 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

%I A238190 #27 Oct 06 2017 08:36:40
%S A238190 1,1,1,1,1,2,1,2,2,1,3,4,1,3,8,1,4,12,3,1,4,18,8,1,5,24,22,1,5,32,40,
%T A238190 6,1,6,40,73,22,1,6,50,112,66,1,7,60,172,146,10,1,7,72,240,292,48,1,8,
%U A238190 84,335,516,174,1,8,98,440,860,448,20
%N A238190 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.
%H A238190 Andrew Howroyd, <a href="/A238190/b238190.txt">Table of n, a(n) for n = 3..974</a>
%H A238190 Christopher Hunt Gribble, <a href="/A238009/a238009_1.cpp.txt">C++ program</a>
%e A238190 The first 13 rows of T(n,k) are:
%e A238190 .\ k    0     1     2     3     4     5
%e A238190 n
%e A238190 3       1     1
%e A238190 4       1     1
%e A238190 5       1     2
%e A238190 6       1     2     2
%e A238190 7       1     3     4
%e A238190 8       1     3     8
%e A238190 9       1     4    12     3
%e A238190 10      1     4    18     8
%e A238190 11      1     5    24    22
%e A238190 12      1     5    32    40     6
%e A238190 13      1     6    40    73    22
%e A238190 14      1     6    50   112    66
%e A238190 15      1     7    60   172   146    10
%t A238190 T[n_, k_] := (2^k Binomial[n - 2k, k] + (Boole[EvenQ[k]] + Boole[OddQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k + 1, 2] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* _Jean-François Alcover_, Oct 06 2017, after _Andrew Howroyd_ *)
%o A238190 (C++) See Gribble link.
%o A238190 (PARI)
%o A238190 T(n,k)={(2^k*binomial(n-2*k,k) + ((k%2==0)+(n%2==1||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-2*k-(n%2))/2,k\2))/4}
%o A238190 for(n=2,20,for(k=0,floor(n/3), print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 29 2017
%Y A238190 Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, 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 A238190 tabf,nonn
%O A238190 3,6
%A A238190 _Christopher Hunt Gribble_, Feb 19 2014
%E A238190 Link to C++ program and xrefs updated by _Christopher Hunt Gribble_, Apr 25 2015
%E A238190 Terms a(51) and beyond from _Andrew Howroyd_, May 29 2017