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A293061
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Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, counting enantiomorphs.
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4
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1, 2, 2, 10, 7, 17, 32, 75, 80, 230, 271, 343, 1091, 1594, 4894, 955
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OFFSET
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0,2
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COMMENTS
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T(n,0) count n-dimensional crystallographic point groups, T(n,n) count n-dimensional space groups (i.e., right border is A006227). The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., symmetry groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional space groups.
The Bohm symbols for these groups are G_{n,k}, except for the case k=n, when it is G_n.
Some groups have their own names:
T(2,1): frieze groups
T(2,2): wallpaper groups
T(3,1): rod groups
T(3,2): layer groups
[Palistrant, 2012, p. 476] gives correct T(4,k), k=0,1,2,3 but incorrect T(4,4). For correct value of T(4,4), see [Souvignier, 2006, p. 80].
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LINKS
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W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
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EXAMPLE
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The triangle begins:
1;
2, 2;
10, 7, 17;
32, 75, 80, 230;
271, 343, 1091, 1594, 4894;
955, ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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