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A231568 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

%I A231568 #37 Oct 06 2017 08:36:26
%S A231568 1,1,1,1,1,2,1,2,1,3,2,1,3,4,1,4,8,1,4,12,1,5,18,3,1,5,24,8,1,6,32,22,
%T A231568 1,6,40,40,1,7,50,73,6,1,7,60,112,22,1,8,72,172,66,1,8,84,240,146,1,9,
%U A231568 98,335,292,10,1,9,112,440,516,48,1,10,128,578,860,174
%N A231568 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.
%H A231568 Andrew Howroyd, <a href="/A231568/b231568.txt">Table of n, a(n) for n = 4..989</a>
%H A231568 Christopher Hunt Gribble, <a href="/A238009/a238009_1.cpp.txt">C++ program</a>
%e A231568 The first 14 rows of T(n,k) are:
%e A231568 .\  k    0      1      2      3     4
%e A231568 n
%e A231568 4        1      1
%e A231568 5        1      1
%e A231568 6        1      2
%e A231568 7        1      2
%e A231568 8        1      3      2
%e A231568 9        1      3      4
%e A231568 10       1      4      8
%e A231568 11       1      4     12
%e A231568 12       1      5     18      3
%e A231568 13       1      5     24      8
%e A231568 14       1      6     32     22
%e A231568 15       1      6     40     40
%e A231568 16       1      7     50     73     6
%e A231568 17       1      7     60    112    22
%t A231568 T[n_, k_] := (2^k Binomial[n - 3k, k] + (Boole[EvenQ[k]] + Boole[EvenQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k+1, 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 A231568 (C++) See Gribble link.
%o A231568 (PARI)
%o A231568 T(n,k)={(2^k*binomial(n-3*k,k) + ((k%2==0)+(n%2==0||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
%o A231568 for(n=2,20,for(k=0,floor(n/4), print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 29 2017
%Y A231568 Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.
%K A231568 tabf,nonn
%O A231568 4,6
%A A231568 _Christopher Hunt Gribble_, Feb 23 2014
%E A231568 Terms extended and xrefs updated by _Christopher Hunt Gribble_, Apr 26 2015
%E A231568 Terms a(32) and beyond from _Andrew Howroyd_, May 29 2017