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Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.
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%I #9 May 29 2017 21:16:25

%S 1,2,1,2,1,4,1,4,1,6,6,1,6,14,1,8,28,1,8,44,1,10,66,20,1,10,90,64,1,

%T 12,120,168,1,12,152,320,1,14,190,572,72,1,14,230,896,328,1,16,276,

%U 1360,984,1,16,324,1920,2264,1,18,378,2660,4528,272

%N Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 7 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

%H Andrew Howroyd, <a href="/A257523/b257523.txt">Table of n, a(n) for n = 4..860</a>

%H Christopher Hunt Gribble, <a href="/A238009/a238009_1.cpp.txt">C++ program</a>

%e The first 9 rows of T(n,k) are:

%e .\ k 0 1 2 3

%e n

%e 4 1 2

%e 5 1 2

%e 6 1 4

%e 7 1 4

%e 8 1 6 6

%e 9 1 6 14

%e 10 1 8 28

%e 11 1 8 44

%e 12 1 10 66 20

%e 13 1 10 90 64

%e 14 1 12 120 168

%e 15 1 12 152 320

%o (PARI)

%o T(n,k)={(4^k*binomial(n-3*k,k) + ((n%2==0||k%2==0)+(k%2==0)+(k==0)) * 4^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}

%o for(n=4,15,for(k=0,(n\4), print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 29 2017

%Y Cf. A034851, A226048, A102541, A226290, A238009, 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 4,2

%A _Christopher Hunt Gribble_, Apr 27 2015

%E Terms a(24) and beyond by _Andrew Howroyd_, May 29 2017