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A238552 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 6 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

%I A238552 #23 Oct 06 2017 08:36:47
%S A238552 1,2,1,2,1,4,1,4,1,6,4,1,6,9,1,8,18,1,8,28,1,10,42,10,1,10,57,28,1,12,
%T A238552 76,76,1,12,96,140,1,14,120,254,25,1,14,145,392,107,1,16,174,600,321,
%U A238552 1,16,204,840,731,1,18,238,1170,1462,70,1,18,273,1540,2610,366
%N A238552 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 6 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.
%H A238552 Andrew Howroyd, <a href="/A238552/b238552.txt">Table of n, a(n) for n = 4..989</a>
%H A238552 Christopher Hunt Gribble, <a href="/A238009/a238009_1.cpp.txt">C++ program</a>
%e A238552 The first 14 rows of T(n,k) are:
%e A238552 .\ k    0      1      2      3     4
%e A238552 n
%e A238552 4       1      2
%e A238552 5       1      2
%e A238552 6       1      4
%e A238552 7       1      4
%e A238552 8       1      6      4
%e A238552 9       1      6      9
%e A238552 10      1      8     18
%e A238552 11      1      8     28
%e A238552 12      1     10     42     10
%e A238552 13      1     10     57     28
%e A238552 14      1     12     76     76
%e A238552 15      1     12     96    140
%e A238552 16      1     14    120    254    25
%e A238552 17      1     14    145    392   107
%t A238552 T[n_, k_] := ((3^k + 1) Binomial[n - 3k, k] + Boole[EvenQ[k] || EvenQ[n]]*(3^Quotient[k + 1, 2] + 3^Quotient[k, 2]) * Binomial[(n - 3k - Mod[k, 2] - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 4, 20}, {k, 0, Floor[n/4]}] // Flatten (* _Jean-François Alcover_, Oct 06 2017, after _Andrew Howroyd_ *)
%o A238552 (C++) See Gribble link.
%o A238552 (PARI)
%o A238552 T(n,k)={((3^k+1)*binomial(n-3*k,k) + (k%2==0||n%2==0) * (3^((k+1)\2)+3^(k\2)) * binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
%o A238552 for(n=4,20,for(k=0,floor(n/4), print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 29 2017
%Y A238552 Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551,  A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.
%K A238552 tabf,nonn
%O A238552 4,2
%A A238552 _Christopher Hunt Gribble_, Feb 28 2014
%E A238552 Terms corrected and xrefs updated by _Christopher Hunt Gribble_, Apr 27 2015
%E A238552 Terms a(28) and beyond from _Andrew Howroyd_, May 29 2017