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%I #17 Feb 21 2024 08:20:09
%S 1,2,2,10,7,17,32,67,80,219,227,343,1076,1594,4783,955
%N Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, not counting enantiomorphs.
%C T(n,0) count n-dimensional crystallographic point groups (i.e., left border is A004028), T(n,n) count n-dimensional space groups (i.e., right border is A004029). 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.
%C The Bohm symbols for these groups are G_{n,k}, except for the case k=n, when it is G_n.
%C Some groups have their own names:
%C T(2,1): frieze groups
%C T(2,2): wallpaper groups
%C T(3,1): rod groups
%C T(3,2): layer groups
%C See [Palistrant, 2012, p. 476] for row 4.
%H M. I. Aroyo et al, <a href="http://www.cryst.ehu.es/">Bilbao Crystallographic Server</a>
%H International Union of Crystallography, <a href="http://it.iucr.org/">International Tables for Crystallography</a>, volumes A and E.
%H A. F. Palistrant, <a href="https://doi.org/10.1134/S1063774512040104">Complete scheme of four-dimensional crystallographic symmetry groups</a>, Crystallography Reports, 57 (2012), 471-477.
%H W. Plesken and T. Schulz, <a href="http://wwwb.math.rwth-aachen.de/carat/">CARAT Homepage</a>
%H W. Plesken and T. Schulz, <a href="/A006226/a006226.pdf">CARAT Homepage</a> [Cached copy in pdf format (without subsidiary pages), with permission]
%H B. Souvignier, <a href="https://doi.org/10.1524/zkri.2006.221.1.77">The four-dimensional magnetic point and space groups</a>, Z. Kristallogr., 221 (2006), 77-82.
%H Wikipedia: <a href="https://en.wikipedia.org/wiki/Space_group">Space group</a>, <a href="https://en.wikipedia.org/wiki/Crystallographic_point_group">Crystallographic point group</a>, <a href="https://en.wikipedia.org/wiki/Line_group">Line group</a>, <a href="https://en.wikipedia.org/wiki/Frieze_group">Frieze group</a>, <a href="https://en.wikipedia.org/wiki/Wallpaper_group">Wallpaper group</a>, <a href="https://en.wikipedia.org/wiki/Rod_group">Rod group</a>, <a href="https://en.wikipedia.org/wiki/Layer_group">Layer group</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%e The triangle begins:
%e 1;
%e 2, 2;
%e 10, 7, 17;
%e 32, 67, 80, 219;
%e 227, 343, 1076, 1594, 4783;
%e 955, ...
%Y Cf. A004028, A004029, A293061, A293062, A293063.
%K nonn,tabl,hard,more
%O 0,2
%A _Andrey Zabolotskiy_, Sep 29 2017
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