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
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).
Programs
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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 *)
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
Extensions
Links corrected by Jan Kristian Haugland, Mar 01 2009
More terms from Colin Barker, May 10 2013
Comments