A038620 Growth function (or coordination sequence) of the infinite cubic graph corresponding to the srs net (a(n) = number of nodes at distance n from a fixed node).

1, 3, 6, 12, 24, 35, 48, 69, 86, 108, 138, 161, 192, 231, 260, 300, 348, 383, 432, 489, 530, 588, 654, 701, 768, 843, 896, 972, 1056, 1115, 1200, 1293, 1358, 1452, 1554, 1625, 1728, 1839, 1916, 2028, 2148, 2231, 2352, 2481, 2570, 2700, 2838, 2933, 3072, 3219

Offset

0,2

Comments

Other names for this structure are triamond, the Laves graph, K_4 lattice, (10,3)-a, or the srs net. A290705 is the theta series of the most symmetric embedding of this graph into space. - Andrey Zabolotskiy, Oct 05 2017

Sunada mentions several other contexts in chemistry and physics where this net occurs. - N. J. A. Sloane, Sep 25 2018

Also, coordination sequence of the hydrogen peroxide lattice. - Sean A. Irvine, May 09 2021

References

A. F. Wells, Three-dimensional Nets and Polyhedra, Wiley, 1977. See the net (10,3)-a.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Thomas Bewley, Paul Belitz, and Joseph Cessna, New horizons in sphere packing theory, part I: fundamental concepts & constructions, from dense to rare. See p. 18, row srs

J. K. Haugland, Classification of certain subgraphs of the 3-dimensional grid, J. Graph Theory, 42 (2003), 34-60.

J. K. Haugland, Illustration

J. K. Haugland, Illustration [Cached copy, with permission] This illustration presents a different (less symmetric) embedding of the srs net into space.

M. O'Keeffe, Coordination sequences for hyperbolic tilings, Zeitschrift für Kristallographie, 213 (1998), 135-140 (see next-to-last table, row 10_5.10_5.10_5).

Reticular Chemistry Structure Resource, srs

Toshikazu Sunada, Crystals that nature might miss creating, Notices Amer. Math. Soc. 55 (No. 2, 2008), 208-215.

Toshikazu Sunada, Correction to "Crystals That Nature Might Miss Creating", Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343.

Toshikazu Sunada, Correction to "Crystals That Nature Might Miss Creating", Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343. [Annotated scanned copy]

Wikipedia, Laves graph

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Formula

a(0)=1, a(1)=3, a(2)=6; for n>=3: if n == 0 (mod 3), a(n) = 4n^2/3; if n == 1 (mod 3), a(n) = (4n^2 + n + 4)/3; if n == 2 (mod 3), a(n) = (4n^2 - n + 10)/3.

G.f.: -(x+1)*(2*x^8-4*x^7+3*x^6-x^5+6*x^4+2*x^3+2*x^2+x+1) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, May 10 2013

Mathematica

CoefficientList[Series[-(x + 1) (2 x^8 - 4 x^7 + 3 x^6 - x^5 + 6 x^4 + 2 x^3 + 2 x^2 + x + 1)/((x - 1)^3 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 3, 6, 12, 24, 35, 48, 69, 86, 108}, 50] (* Harvey P. Dale, Sep 02 2017 *)

Crossrefs

Cf. A038621 (partial sums), A290705 (theta series).

Sequence in context: A316318 A173216 A003204 * A250300 A363692 A330132

Adjacent sequences: A038617 A038618 A038619 * A038621 A038622 A038623

Keyword

nonn,easy

Author

Jan Kristian Haugland

Extensions

Links corrected by Jan Kristian Haugland, Mar 01 2009

More terms from Colin Barker, May 10 2013

Status

approved