A238190 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 4 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 1, 3, 8, 1, 4, 12, 3, 1, 4, 18, 8, 1, 5, 24, 22, 1, 5, 32, 40, 6, 1, 6, 40, 73, 22, 1, 6, 50, 112, 66, 1, 7, 60, 172, 146, 10, 1, 7, 72, 240, 292, 48, 1, 8, 84, 335, 516, 174, 1, 8, 98, 440, 860, 448, 20

Offset: 3

Author

Christopher Hunt Gribble, Feb 19 2014

Keywords

• tabf
• nonn

Examples

			The first 13 rows of T(n,k) are:
.\ k    0     1     2     3     4     5
n
3       1     1
4       1     1
5       1     2
6       1     2     2
7       1     3     4
8       1     3     8
9       1     4    12     3
10      1     4    18     8
11      1     5    24    22
12      1     5    32    40     6
13      1     6    40    73    22
14      1     6    50   112    66
15      1     7    60   172   146    10
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..974
• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Programs

Mathematica

T[n_, k_] := (2^k Binomial[n - 2k, k] + (Boole[EvenQ[k]] + Boole[OddQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k + 1, 2] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

PARI

T(n,k)={(2^k*binomial(n-2*k,k) + ((k%2==0)+(n%2==1||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-2*k-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/3), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Extensions

Link to C++ program and xrefs updated by Christopher Hunt Gribble, Apr 25 2015

Terms a(51) and beyond from Andrew Howroyd, May 29 2017