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A232440 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5), read by rows.

%I A232440 #31 Oct 06 2017 08:36:33
%S A232440 1,1,1,1,1,2,1,2,1,3,1,3,1,1,4,2,1,4,4,1,5,6,1,5,9,1,6,12,1,1,6,16,2,
%T A232440 1,7,20,6,1,7,25,10,1,8,30,19,1,8,36,28,1,1,9,42,44,3,1,9,49,60,9,1,
%U A232440 10,56,85,19,1,10,64,110,38,1,11,72,146,66,1
%N A232440 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5), read by rows.
%H A232440 Andrew Howroyd, <a href="/A232440/b232440.txt">Table of n, a(n) for n = 5..989</a>
%H A232440 Christopher Hunt Gribble, <a href="/A238009/a238009_1.cpp.txt">C++ program</a>
%e A232440 The first 9 rows of T(n,k) are:
%e A232440 .\ k    0      1      2
%e A232440 n
%e A232440 5       1      1
%e A232440 6       1      1
%e A232440 7       1      2
%e A232440 8       1      2
%e A232440 9       1      3
%e A232440 10      1      3      1
%e A232440 11      1      4      2
%e A232440 12      1      4      4
%e A232440 13      1      5      6
%t A232440 T[n_, k_] := (Binomial[n - 4k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 4k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 5, 20}, {k, 0, Quotient[n, 5]}] // Flatten (* _Jean-François Alcover_, Oct 06 2017, after _Andrew Howroyd_ *)
%o A232440 (C++) See Gribble link.
%o A232440 (PARI)
%o A232440 T(n,k)={(binomial(n-4*k,k) + (k%2==0||n%2==1)*binomial((n-4*k-n%2)/2,k\2))/2}
%o A232440 for(n=5,20,for(k=0,(n\5), print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 29 2017
%Y A232440 Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.
%K A232440 tabf,nonn
%O A232440 5,6
%A A232440 _Christopher Hunt Gribble_, Feb 23 2014
%E A232440 Terms extended and xrefs updated by _Christopher Hunt Gribble_, Apr 26 2015
%E A232440 Terms a(27) and beyond from _Andrew Howroyd_, May 29 2017