A006768 -id:A006768

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A049430

Triangle read by rows: T(n,d) is the number of distinct properly d-dimensional polyominoes (or polycubes) with n cells (n >= 1, d >= 0).

+10
19

1, 0, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 11, 3, 0, 1, 34, 77, 35, 6, 0, 1, 107, 499, 412, 104, 11, 0, 1, 368, 3442, 4888, 2009, 319, 23, 0, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 0, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 0, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

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

OFFSET

1,9

LINKS

Table of n, a(n) for n=1..66.

W. F. Lunnon, Counting multidimensional polyominoes. Computer Journal 18 (1975), no. 4, pp. 366-367.

EXAMPLE

Triangle begins:

0 1

0 1 1

0 1 4 2

0 1 11 11 3

0 1 34 77 35 6

0 1 107 499 412 104 11

0 1 368 3442 4888 2009 319 23

0 1 1284 24128 57122 36585 8869 951 47

0 1 4654 173428 667959 647680 231574 36988 2862 106

0 1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235

...

CROSSREFS

Cf. A049429 (col. d=0 omitted), A195738 (oriented), A195739 (fixed).

Row sums give A005519. Columns give A006765, A006766, A006767, A006768.

Diagonals (with algorithms) are A000055, A036364, A355053.

Cf. A330891 (cumulative sums of the rows).

KEYWORD

nonn,nice,tabl,hard

AUTHOR

Richard C. Schroeppel

EXTENSIONS

Edited by N. J. A. Sloane, Sep 23 2011

More terms from John Niss Hansen, Mar 31 2015

STATUS

approved

A049429

Triangle T(n,d) = number of distinct d-dimensional polyominoes (or polycubes) with n cells (0 < d < n).

+10
11

1, 1, 1, 1, 4, 2, 1, 11, 11, 3, 1, 34, 77, 35, 6, 1, 107, 499, 412, 104, 11, 1, 368, 3442, 4888, 2009, 319, 23, 1, 1284, 24128, 57122, 36585, 8869, 951, 47, 1, 4654, 173428, 667959, 647680, 231574, 36988, 2862, 106, 1, 17072, 1262464, 7799183, 11173880, 5712765, 1297366, 146578, 8516, 235

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

OFFSET

2,5

COMMENTS

These are unoriented polyominoes of the regular tilings with Schläfli symbols {oo}, {4,4}, {4,3,4}, {4,3,3,4}, etc. Each tile is a regular orthotope (hypercube). For unoriented polyominoes, chiral pairs are counted as one. The dimension of the convex hull of the cell centers determines the dimension d. - Robert A. Russell, Aug 09 2022

REFERENCES

Dan Hoey, Bill Gosper and Richard C. Schroeppel, Discussions in Math-Fun Mailing list, circa Jul 13 1999.

LINKS

Table of n, a(n) for n=2..56.

R. C. Schroeppel, A few mathematical experiments

EXAMPLE

From Robert A. Russell, Aug 09 2022: (Start)

Triangle begins with T(2,1):

n\d 1 2 3 4 5 6 7 8 9 10

2 1

3 1 1

4 1 4 2

5 1 11 11 3

6 1 34 77 35 6

7 1 107 499 412 104 11

8 1 368 3442 4888 2009 319 23

9 1 1284 24128 57122 36585 8869 951 47

10 1 4654 173428 667959 647680 231574 36988 2862 106

11 1 17072 1262464 7799183 11173880 5712765 1297366 146578 8516 235

(End)