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Low-dimensional lattices. VII: Coordination sequences. (English) Zbl 1066.11505

Summary: The coordination sequence \(\{S(n) \}\) of a lattice or net gives the number of nodes that are \(n\) bonds away from a given node. \(S(1)\) is the familiar coordination number. Extending the work of O’Keeffe and others, we give explicit formulae for the coordination sequences of the root lattices \(A_d\), \(D_d\), \(E_6\), \(E_7\), \(E_8\) and their duals. Proofs are given for many of the formulae and for the fact that, in every case, \(S(n)\) is a polynomial in \(n\), although some of the individual formulae are conjectural. In the majority of cases, the set of nodes that are at most \(n\) bonds away from a given node form a polytopal cluster whose shape is the same as that of the contact polytope for the lattice. It is also shown that among all the Barlow packings in three dimensions the hexagonal close packing has the greatest coordination sequence, and the face-centred cubic lattice the smallest, as conjectured by O’Keeffe.
Part VI, cf. ibid. 436, No. 1896, 55–68 (1992; Zbl 0747.11027).

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
11H31 Lattice packing and covering (number-theoretic aspects)

Citations:

Zbl 0747.11027
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Full Text: DOI

Online Encyclopedia of Integer Sequences:

Number of points in interior of n-th crystal ball in E_8 lattice.
Centered octahedral numbers (crystal ball sequence for cubic lattice).
Centered 4-dimensional orthoplex numbers (crystal ball sequence for 4-dimensional cubic lattice).
Crystal ball sequence for 5-dimensional cubic lattice.
Crystal ball sequence for 6-dimensional cubic lattice.
Crystal ball sequence for 7-dimensional cubic lattice.
Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice).
Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).
Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2,
Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequence for f.c.c. or A_3 or D_3 lattice.
Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.
Crystal ball sequence for hexagonal close-packing.
Crystal ball sequence for D_4 lattice.
Coordination sequence for hexagonal close-packing.
Coordination sequence for D_4 lattice.
Crystal ball sequence for diamond.
Coordination sequence for diamond.
Number of orbits on points that are at n steps from 0 in E_8 lattice.
Coordination sequence for E_8 lattice.
Crystal ball sequence for E_8 lattice.
Number of orbits of norm 2n vectors in E_8 lattice.
Coordination sequence for D_5 lattice.
Crystal ball sequence for D_5 lattice.
Coordination sequence for D_6 lattice.
Crystal ball sequence for D_6 lattice.
Coordination sequence for D_7 lattice.
Crystal ball sequence for D_7 lattice.
Coordination sequence for D_8 lattice.
Crystal ball sequence for D_8 lattice.
Number of orbits on points that are at n steps from the origin in the f.c.c. lattice.
Number of orbits on points that are at n steps from 0 in D_4 lattice.
Number of orbits on points that are at n steps from 0 in D_5 lattice.
Number of orbits on points that are at n steps from 0 in D_6 lattice.
Number of orbits on points that are at n steps from 0 in D_7 lattice.
Number of orbits on points that are at n steps from 0 in D_8 lattice.
Number of orbits on points that are at n steps from 0 in D_9 lattice.
Number of orbits on points that are at n steps from 0 in D_10 lattice.
Coordination sequence for D_9 lattice.
Crystal ball sequence for D_9 lattice.
Coordination sequence for D_10 lattice.
Crystal ball sequence for D_10 lattice.
Coordination sequence for A_4 lattice.
Crystal ball sequence for A_4 lattice.
Coordination sequence for A_5 lattice.
Crystal ball sequence for A_5 lattice.
Coordination sequence for A_6 lattice.
Crystal ball sequence for A_6 lattice.
Coordination sequence for A_7 lattice.
Crystal ball sequence for A_7 lattice.
Coordination sequence for A_8 lattice.
Crystal ball sequence for A_8 lattice.
Coordination sequence for A_9 lattice.
Crystal ball sequence for A_9 lattice.
Coordination sequence for A_10 lattice.
Crystal ball sequence for A_10 lattice.
Coordination sequence for E_7 lattice.
Crystal ball sequence for E_7 lattice.
Coordination sequence for E_6 lattice.
Crystal ball sequence for E_6 lattice.
Coordination sequence for {E_6}* lattice.
Crystal ball sequence for {E_6}* lattice.
Coordination sequence for 4-dimensional cubic lattice (points on surface of 4-dimensional cross-polytope).
Coordination sequence for 5-dimensional cubic lattice.
Coordination sequence for 6-dimensional cubic lattice.
Coordination sequence for 7-dimensional cubic lattice.
Coordination sequence for 8-dimensional cubic lattice.
Crystal ball sequence for 8-dimensional cubic lattice.
Coordination sequence for 9-dimensional cubic lattice.
Crystal ball sequence for 9-dimensional cubic lattice.
Coordination sequence for 10-dimensional cubic lattice.
Crystal ball sequence for 10-dimensional cubic lattice.
Coordination sequence for hexagonal lattice.
Square the entries of Pascal’s triangle.
Expansion of (1-x^5) / (1-x)^5.
Coordination sequence for {A_6}* lattice.
Triangle of truncated triangular numbers: k-th term in n-th row is number of dots in hexagon of sides k, n-k, k, n-k, k, n-k.
Truncated triangular numbers (of form n*(n-3)/2 - k^2+k*n+1 for 1<=k<n).
Coordination sequence for lattice {E_7}*.
Crystal ball sequence for lattice {E_7}*.
Coordination sequence for squashed {D_5}* lattice, perhaps the smallest example of a ”non-superficial” lattice.
Crystal ball sequence for squashed {D_5}^* lattice, perhaps the smallest example of a ”non-superficial” lattice.
Number of points of L1 norm 2n in root system version of E_8 lattice.
Simon Plouffe’s conjectured extension of sequence A008368.
Number of points of L1 norm 4 in cubic lattice Z^n.
Number of points of L1 norm 5 in cubic lattice Z^n.
Number of points of L1 norm 6 in cubic lattice Z^n.
Number of points of L1 norm 7 in cubic lattice Z^n.
Number of points of L1 norm 8 in cubic lattice Z^n.
Number of points of L1 norm 9 in cubic lattice Z^n.
Number of points of L1 norm 10 in cubic lattice Z^n.
Number of points of L1 norm 11 in cubic lattice Z^n.
Number of points of L1 norm 12 in cubic lattice Z^n.
Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
Coordination sequence for 11-dimensional cubic lattice.
Coordination sequence for 12-dimensional cubic lattice.
Coordination sequence for 13-dimensional cubic lattice.
Coordination sequence for 14-dimensional cubic lattice.
Coordination sequence for 15-dimensional cubic lattice.
Coordination sequence for 16-dimensional cubic lattice.
Coordination sequence for 17-dimensional cubic lattice.
Coordination sequence for 18-dimensional cubic lattice.
Coordination sequence for 19-dimensional cubic lattice.
Coordination sequence for 20-dimensional cubic lattice.
Coordination sequence for 21-dimensional cubic lattice.
Coordination sequence for 22-dimensional cubic lattice.
Coordination sequence for 23-dimensional cubic lattice.
Coordination sequence for 24-dimensional cubic lattice.
Coordination sequence for 25-dimensional cubic lattice.
Coordination sequence for 26-dimensional cubic lattice.
Coordination sequence for 27-dimensional cubic lattice.
Coordination sequence for 28-dimensional cubic lattice.
Coordination sequence for 29-dimensional cubic lattice.
Coordination sequence for 30-dimensional cubic lattice.
Coordination sequence for 31-dimensional cubic lattice.
Coordination sequence for 32-dimensional cubic lattice.
Coordination sequence for 33-dimensional cubic lattice.
Coordination sequence for 34-dimensional cubic lattice.
Coordination sequence for 35-dimensional cubic lattice.
Coordination sequence for 36-dimensional cubic lattice.
Coordination sequence for 37-dimensional cubic lattice.
Coordination sequence for 38-dimensional cubic lattice.
Coordination sequence for 39-dimensional cubic lattice.
Coordination sequence for 40-dimensional cubic lattice.
Coordination sequence for 41-dimensional cubic lattice.
Coordination sequence for 42-dimensional cubic lattice.
Coordination sequence for 43-dimensional cubic lattice.
Coordination sequence for 44-dimensional cubic lattice.
Coordination sequence for 45-dimensional cubic lattice.
Coordination sequence for 46-dimensional cubic lattice.
Coordination sequence for 47-dimensional cubic lattice.
Coordination sequence for 48-dimensional cubic lattice.
Coordination sequence for 49-dimensional cubic lattice.
Coordination sequence for 50-dimensional cubic lattice.
Coordination sequence for C_9 lattice.
Coordination sequence for C_10 lattice.
Coordination sequence for C_11 lattice.
Coordination sequence for C_12 lattice.
Coordination sequence for C_13 lattice.
Coordination sequence for C_14 lattice.
Coordination sequence for C_15 lattice.
Coordination sequence for C_16 lattice.
Coordination sequence for C_17 lattice.
Coordination sequence for C_18 lattice.
Coordination sequence for C_19 lattice.
Coordination sequence for C_20 lattice.
Coordination sequence for C_21 lattice.
Coordination sequence for C_22 lattice.
Coordination sequence for C_23 lattice.
Coordination sequence for C_24 lattice.
Coordination sequence for C_25 lattice.
Coordination sequence for C_26 lattice.
Coordination sequence for C_27 lattice.
Coordination sequence for C_28 lattice.
Coordination sequence for C_29 lattice.
Coordination sequence for C_30 lattice.
Coordination sequence for C_31 lattice.
Coordination sequence for C_32 lattice.
Coordination sequence for C_33 lattice.
Coordination sequence for C_34 lattice.
Coordination sequence for C_35 lattice.
Coordination sequence for C_36 lattice.
Coordination sequence for C_37 lattice.
Coordination sequence for C_38 lattice.
Coordination sequence for C_39 lattice.
Coordination sequence for C_40 lattice.
Coordination sequence for C_41 lattice.
Coordination sequence for C_42 lattice.
Coordination sequence for C_43 lattice.
Coordination sequence for C_44 lattice.
Coordination sequence for C_45 lattice.
Coordination sequence for C_46 lattice.
Coordination sequence for C_47 lattice.
Coordination sequence for C_48 lattice.
Coordination sequence for C_49 lattice.
Coordination sequence for C_50 lattice.
Coordination sequence for A_11 lattice.
Coordination sequence for A_12 lattice.
Coordination sequence for A_13 lattice.
Coordination sequence for A_14 lattice.
Coordination sequence for A_15 lattice.
Coordination sequence for A_16 lattice.
Coordination sequence for A_17 lattice.
Coordination sequence for A_18 lattice.
Coordination sequence for A_19 lattice.
Coordination sequence for A_20 lattice.
Coordination sequence for A_21 lattice.
Coordination sequence for A_22 lattice.
Coordination sequence for A_23 lattice.
Coordination sequence for A_24 lattice.
Coordination sequence for A_25 lattice.
Coordination sequence for A_26 lattice.
Coordination sequence for A_27 lattice.
Coordination sequence for A_28 lattice.
Coordination sequence for A_29 lattice.
Coordination sequence for A_30 lattice.
Coordination sequence for A_31 lattice.
Coordination sequence for A_32 lattice.
Coordination sequence for A_33 lattice.
Coordination sequence for A_34 lattice.
Coordination sequence for A_35 lattice.
Coordination sequence for A_36 lattice.
Coordination sequence for A_37 lattice.
Coordination sequence for A_38 lattice.
Coordination sequence for A_39 lattice.
Coordination sequence for A_40 lattice.
Coordination sequence for A_41 lattice.
Coordination sequence for A_42 lattice.
Coordination sequence for A_43 lattice.
Coordination sequence for A_44 lattice.
Coordination sequence for A_45 lattice.
Coordination sequence for A_46 lattice.
Coordination sequence for A_47 lattice.
Coordination sequence for A_48 lattice.
Coordination sequence for A_49 lattice.
Coordination sequence for A_50 lattice.
Number of points of l_1 norm n in the ”diamond” lattice D^+_2, i. e. the rectangular lattice generated by vectors (1, 1) and (-1/2, 1/2).
Number of points of l_1 norm n in the ”diamond” lattice D^+_4.
Coordination sequence for diamond structure D^+_6. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_10. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_12. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_14. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_16. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_18. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_20. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_22. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_24. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_26. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_28. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_30. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_32. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_34. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_36. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_38. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_40. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_42. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_44. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_46. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_48. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_50. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_52. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_54. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_56. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_58. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_60. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_62. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_64. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_66. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_68. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_70. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_72. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_74. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_76. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_78. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_80. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_82. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_84. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_86. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_88. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_90. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_92. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_94. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_96. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_98. (Edges defined by l_1 norm = 1.)
Coordination sequence for diamond structure D^+_100. (Edges defined by l_1 norm = 1.)
Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.
Square array T(n,k) read by antidiagonals: coordination sequence for lattice B_n.
Square array T(n,k) read by antidiagonals: coordination sequence for lattice C_n.
Square array T(n,k) read by antidiagonals: coordination sequence for lattice D_n.
Symmetric triangle, read by rows, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice, for n>=0.
Array of coordination sequences for cubic lattices (rows) and of numbers of L1 forms in cubic lattices (columns) (array read by antidiagonals).
Square array, read by ascending antidiagonals, of the crystal ball sequences for the root lattices of type C_n.
Square array, read by antidiagonals, where row n equals the crystal ball sequence for the 2*n-dimensional lattice A_n x A_n.
The number of integer sequences of length d = 2n+1 such that the sum of the terms is 0 and the sum of the absolute values of the terms is d-1.
Triangle read by rows: coordinator triangle for lattice A*_n.
a(n) = binomial(4*n, 2*n) - 4*n*binomial(2*n-2, n-1).
Coordination sequence for Leech lattice.