A231473 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

1, 2, 1, 2, 1, 4, 1, 4, 4, 1, 6, 9, 1, 6, 18, 1, 8, 28, 10, 1, 8, 42, 28, 1, 10, 57, 76, 1, 10, 76, 140, 25, 1, 12, 96, 254, 107, 1, 12, 120, 392, 321, 1, 14, 145, 600, 731, 70, 1, 14, 174, 840, 1462, 366, 1, 16, 204, 1170, 2610, 1308, 1, 16, 238, 1540, 4350, 3416, 196

Offset

3,2

Links

Andrew Howroyd, Table of n, a(n) for n = 3..974

Christopher Hunt Gribble, C++ program

Example

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

Mathematica

T[n_, k_] := ((3^k + 1)*Binomial[n - 2k, k] + Boole[EvenQ[k] || OddQ[n]]*(3^(Quotient[(k + 1), 2]) + 3^Quotient[k, 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 *)

Prog

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

Crossrefs

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

Sequence in context: A325677 A343411 A287477 * A216652 A331980 A055684

Adjacent sequences: A231470 A231471 A231472 * A231474 A231475 A231476

Keyword

tabf,nonn

Author

Christopher Hunt Gribble, Feb 23 2014

Extensions

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

Terms a(40) and beyond from Andrew Howroyd, May 29 2017

Status

approved