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A300812 Irregular triangle T(n,c) read by rows: the number of clusters of n spheres centered on f.c.c. lattice sites with c contacts.

%I A300812 #20 Oct 10 2022 10:36:02
%S A300812 1,0,1,0,0,3,1,0,0,0,13,4,2,1,0,0,0,0,75,35,16,3,2,0,0,0,0,0,557,384,
%T A300812 184,54,24,5,2,1,0,0,0,0,0,0,4808,4230,2354,834,355,104,37,9,2,1,0,0,
%U A300812 0,0,0,0,0,44334,47328,30517,13081,5716,2083,749,253,70,20,4,3
%N A300812 Irregular triangle T(n,c) read by rows: the number of clusters of n spheres centered on f.c.c. lattice sites with c contacts.
%C A300812 Each of the n spheres is centered at a different vertex of the face-centered-cubic lattice.
%C A300812 "Clusters of spheres" means that the (simple, undirected) graph, which has edges for each "bond" where two spheres touch, is connected. The number of contacts equals the number of edges in the graph.
%C A300812 T(n,c) counts "free" clusters, which means that clusters are considered distinct if they cannot be mapped onto each other by one of the 48 symmetries of the octahedral group plus shifts along the directions of the face centers.
%C A300812 Since there are 12 nearest neighbors in the f.c.c. lattice, the degree of each vertex in the contact graph is in the interval [1,12].
%C A300812 T(n,c) = 0 for c < n: the minimum number of contacts is realized with chains of spheres, where the two spheres at the ends have degree 1 and the others degree 2.
%C A300812 The row length of row n of the triangle should equal 1+A214813(n) ... but does not for n=5, see the comment in A214813.
%H A300812 R. J. Mathar, <a href="/A300812/a300812.pdf">Illustration for geometries of clusters of 4 and 5 atoms</a>(2018)
%e A300812 The values T(n,c) start with n=1 sphere for 0 <= c contacts as:
%e A300812   1
%e A300812   0 1
%e A300812   0 0 3 1
%e A300812   0 0 0 13 4 2 1
%e A300812   0 0 0 0 75 35 16 3 2
%e A300812   0 0 0 0 0 557 384 184 54 24 5 2 1
%e A300812   0 0 0 0 0 0 4808 4230 2354 834 355 104 37 9 2 1
%e A300812   0 0 0 0 0 0 0 44334 47328 30517 13081 5716 2083 749 253 70 20 4 3
%e A300812 T(2,1) = 1 because there is one cluster with two spheres which touch each other at one point: (0,0,0), (1/2,0,1/2).
%e A300812 T(3,2) = 3 counts three spheres in three different geometries: (i) (0,0,0), (1/2,1/2,0), (1,0,1), linear, (ii) (0,0,0), (1/2,1/2,0), (1,1,0) with 90-degree bond angle, (iii) (0,0,0), (1/2,1/2,0), (1,1/2,1/2) with 120-degree bond angle.
%e A300812 T(3,3) = 1 counts the planar triangular canonball base arrangement: (0,0,0), (1/2,1/2,0), (1/2,0,1/2).
%Y A300812 Cf. A038173 (row sums), A214813.
%K A300812 nonn,tabf
%O A300812 1,6
%A A300812 _R. J. Mathar_, Mar 13 2018