A008579 Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane.

1, 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, 54, 68, 62, 78, 70, 88, 78, 98, 86, 108, 94, 118, 102, 128, 110, 138, 118, 148, 126, 158, 134, 168, 142, 178, 150, 188, 158, 198, 166, 208, 174, 218, 182, 228, 190, 238, 198, 248, 206, 258, 214, 268, 222, 278

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Interesting because coefficients never become monotonic.

Also the coordination sequence for a planar net made of densely packed circles. - Yuriy Sibirmovsky, Sep 11 2016

Described by J.-G. Eon (2014) as the coordination sequence of the Kagome net. - N. J. A. Sloane, Jan 03 2018

References

• P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 161 (but beware errors).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Pierre de La Harpe, and P. I. Grigorchuk, Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook, Algebra and Logic 37.6 (1998): 353-356.
• Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See p. 51.
• Jean-Guillaume Eon, Topological density of nets: a direct calculation, Acta Crystallographica Section A (Foundations of Crystallography), A60 (2014), 7-18; DOI: 10.1107/S0108767303022037. See Section 5.
• Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 4.
• Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
• Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
• Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
• Tom Karzes, Tiling Coordination Sequences
• Reticular Chemistry Structure Resource, kgm
• Yuriy Sibirmovsky, Illustration of initial terms with densely packed circles.
• N. J. A. Sloane, Illustration of initial terms
• N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Cf. A017137, A017365.

Programs

Haskell

a008579 0 = 1
a008579 1 = 4
a008579 n = (10 - 2*m) * n' + 8*m - 2 where (n',m) = divMod n 2
a008579_list = 1 : 4 : concatMap (\x -> map (* 2) [5*x-1,4*x+3]) [1..]
-- Reinhard Zumkeller, Nov 12 2012

Maple

f := n->if n mod 2 = 0 then 10*(n/2)-2 else 8*(n-1)/2+6 fi;

Mathematica

a[n_?EvenQ] := 10*n/2-2; a[n_?OddQ] := 8*(n-1)/2+6; a[0] = 1; a[1] = 4; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Nov 18 2011, after Maple *)
CoefficientList[Series[(1+2x)(1+2x+2x^2+2x^3-x^4)/(1-x^2)^2,{x,0,50}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,4,8,14,18,22},50] (* Harvey P. Dale, Sep 05 2018 *)

Formula

G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4)/(1 - x^2)^2.

From R. J. Mathar, Nov 26 2014: (Start)

a(2n) = A017365(n), n > 0.

a(2n+1) = A017137(n), n > 0. (End)

From Stefano Spezia, Aug 07 2022: (Start)

a(n) = (9 + (-1)^n)*n/2 - 2*(-1)^n for n > 1.

E.g.f.: 3 - 2*x + (4*x - 2)*cosh(x) + (5*x + 2)*sinh(x). (End)