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.

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, 76, 76, 1, 12, 96, 140, 1, 14, 120, 254, 25, 1, 14, 145, 392, 107, 1, 16, 174, 600, 321, 1, 16, 204, 840, 731, 1, 18, 238, 1170, 1462, 70, 1, 18, 273, 1540, 2610, 366

Offset: 4

Author

Christopher Hunt Gribble, Feb 28 2014

Keywords

• tabf
• nonn

Examples

			The first 14 rows of T(n,k) are:
.\ k    0      1      2      3     4
n
4       1      2
5       1      2
6       1      4
7       1      4
8       1      6      4
9       1      6      9
10      1      8     18
11      1      8     28
12      1     10     42     10
13      1     10     57     28
14      1     12     76     76
15      1     12     96    140
16      1     14    120    254    25
17      1     14    145    392   107
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 4..989
• Christopher Hunt Gribble, C++ program

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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}
for(n=4,20,for(k=0,floor(n/4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 27 2015

Terms a(28) and beyond from Andrew Howroyd, May 29 2017