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

4,2

Links

Andrew Howroyd, Table of n, a(n) for n = 4..989

Christopher Hunt Gribble, C++ program

Example

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

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 *)

Prog

(C++) See Gribble link.
(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

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.

Sequence in context: A193787 A072614 A287597 * A257523 A067044 A325677

Adjacent sequences: A238549 A238550 A238551 * A238553 A238554 A238555

Keyword

tabf,nonn

Author

Christopher Hunt Gribble, Feb 28 2014

Extensions

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

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

Status

approved