A238552 - OEIS

%I #23 Oct 06 2017 08:36:47

%S 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 76,76,1,12,96,140,1,14,120,254,25,1,14,145,392,107,1,16,174,600,321,

%U 1,16,204,840,731,1,18,238,1170,1462,70,1,18,273,1540,2610,366

%N 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 Andrew Howroyd, <a href="/A238552/b238552.txt">Table of n, a(n) for n = 4..989</a>

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

%e The first 14 rows of T(n,k) are:

%e .\ k    0      1      2      3     4

%e n

%e 4       1      2

%e 5       1      2

%e 6       1      4

%e 7       1      4

%e 8       1      6      4

%e 9       1      6      9

%e 10      1      8     18

%e 11      1      8     28

%e 12      1     10     42     10

%e 13      1     10     57     28

%e 14      1     12     76     76

%e 15      1     12     96    140

%e 16      1     14    120    254    25

%e 17      1     14    145    392   107

%t 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 (C++) See Gribble link.

%o (PARI)

%o 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 for(n=4,20,for(k=0,floor(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,  A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

%K tabf,nonn

%O 4,2

%A _Christopher Hunt Gribble_, Feb 28 2014

%E Terms corrected and xrefs updated by _Christopher Hunt Gribble_, Apr 27 2015

%E Terms a(28) and beyond from _Andrew Howroyd_, May 29 2017