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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Links corrected by Jan Kristian Haugland, Mar 01 2009
More terms from Colin Barker, May 10 2013
STATUS
approved