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A224850 Number T(n,k) of tilings of an n X k rectangle using integer sided square tiles reduced for symmetry, where the orbits under the symmetry group of the rectangle, D2, have 1 element; triangle T(n,k), k>=1, 0<=n<k, read by columns. 4
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 3, 1, 1, 5, 2, 12, 6, 1, 1, 3, 3, 5, 7, 17, 1, 1, 8, 3, 25, 11, 106, 44 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

The triangle is:

n\k  1   2   3   4   5   6   7   8 ...

.

0    1   1   1   1   1   1   1   1 ...

1        1   1   1   1   1   1   1 ...

2            1   3   2   5   3   8 ...

3                1   2   2   3   3 ...

4                    3  12   5  25 ...

5                        6   7  11 ...

6                           17 106 ...

7                               44 ...

It appears that sequence T(2,k) consists of 2 interspersed Fibonacci sequences.

The diagonal T(n,n) is A006081. - M. F. Hasler, Jul 25 2013

LINKS

Table of n, a(n) for n=1..36.

Christopher Hunt Gribble, C++ program

FORMULA

T(n,k) + A224861(n,k) + A224867(n,k) = A227690(n,k).

1*T(n,k) + 2*A224861(n,k) + 4*A224867(n,k) = A219924(n,k).

EXAMPLE

T(3,5) = 2 because there are 2 different tilings of the 3 X 5 rectangle by integer sided squares, where any sequence of group D2 operations will only transform each tiling into itself.  Group D2 operations are:

.   the identity operation

.   rotation by 180 degrees

.   reflection about a horizontal axis through the center

.   reflection about a vertical axis through the center

The tilings are:

._________.    ._________.

|_|_|_|_|_|    |_|     |_|

|_|_|_|_|_|    |_|     |_|

|_|_|_|_|_|    |_|_____|_|

CROSSREFS

Cf. A219924, A224697, A227690.

Sequence in context: A136148 A178252 A325498 * A269976 A132409 A030337

Adjacent sequences:  A224847 A224848 A224849 * A224851 A224852 A224853

KEYWORD

nonn,tabl

AUTHOR

Christopher Hunt Gribble, Jul 22 2013

STATUS

approved

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Last modified October 15 00:14 EDT 2019. Contains 328025 sequences. (Running on oeis4.)