A228267 - OEIS

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A228267

Number T(n,k,r) of dissections of an n X k X r rectangular cuboid into integer-sided cubes including rotations and reflections; irregular triangle T(n,k,r), n >= k >= r >= 1 read by rows.

1, 1, 1, 2, 1, 1, 3, 1, 5, 10, 1, 1, 5, 1, 11, 31, 1, 35, 167, 2098, 1, 1, 8, 1, 21, 76, 1, 93, 635, 15511, 1, 314, 3354, 185473, 4006722, 1, 1, 13, 1, 43, 210, 1, 269, 2887, 151378, 1, 1213, 22478, 3243515, 143662050, 1, 6427, 235150, 112411358

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

The main diagonal T(n,n,n) is 1, 2, 10, 2098, 4006722, .... - R. J. Mathar and Rob Pratt, Nov 27 2017

LINKS

Table of n, a(n) for n=1..54.

Christopher Hunt Gribble, C++ program

FORMULA

T(1,1,r) = T(n,n,1) = 1. - R. J. Mathar, Dec 03 2017

T(2,2,r) = A000045(r+1). - R. J. Mathar, Dec 03 2017

T(3,3,r>=1) = 1, 5, 10, 31, ... with g.f. 1/(1-x-4*x^2-x^3). - R. J. Mathar, Dec 03 2017

T(4,4,r>=1) = 1, 35, 167, 2098, 15511, 151378, 1272179, 11574563, 100928230, 900224006, ... with TBD rational g.f. - R. J. Mathar, Dec 03 2017

T(n,n,2) = A063443(n). - R. J. Mathar, Dec 03 2017

EXAMPLE

The irregular triangle begins:

. r 1 2 3 4 ...

n,k

1,1 1

2,1 1

2,2 1 2

3,1 1

3,2 1 3

3,3 1 5 10

4,1 1

4,2 1 5

4,3 1 11 31

4,4 1 35 167 2098

5,1 1

5,2 1 8

5,3 1 21 76

5,4 1 93 635 15511

5,5 1 314 3354 185473 ...

...

T(3,2,2) = 3 because there are 3 distinct dissections of a 3 X 2 X 2 rectangular cuboid into integer-sided cubes. The dissections expanded into 2 dimensions are:

._____. ._____. ._____.

|_|_|_| |_|_|_| |_|_|_|

._____. ._____. ._____.

| |_| | |_| | |_|

|___|_| |___|_| |___|_|

._____. ._____. ._____.

|_| | |_| | |_| |

|_|___| |_|___| |_|___|

CROSSREFS

Cf. A219924.

Sequence in context: A327981 A348447 A277606 * A170820 A339615 A003687

Adjacent sequences: A228264 A228265 A228266 * A228268 A228269 A228270