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

0,2

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).

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)

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 *)

Prog

(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

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.

Sequence in context: A312502 A312503 A312504 * A312505 A312506 A312507

Adjacent sequences: A008576 A008577 A008578 * A008580 A008581 A008582

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved