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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).

%I A038620 #67 Mar 21 2025 13:25:28
%S A038620 1,3,6,12,24,35,48,69,86,108,138,161,192,231,260,300,348,383,432,489,
%T A038620 530,588,654,701,768,843,896,972,1056,1115,1200,1293,1358,1452,1554,
%U A038620 1625,1728,1839,1916,2028,2148,2231,2352,2481,2570,2700,2838,2933,3072,3219
%N 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).
%C A038620 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
%C A038620 Sunada mentions several other contexts in chemistry and physics where this net occurs. - _N. J. A. Sloane_, Sep 25 2018
%C A038620 Also, coordination sequence of the hydrogen peroxide lattice. - _Sean A. Irvine_, May 09 2021
%D A038620 A. F. Wells, Three-dimensional Nets and Polyhedra, Wiley, 1977. See the net (10,3)-a.
%H A038620 Vincenzo Librandi, <a href="/A038620/b038620.txt">Table of n, a(n) for n = 0..1000</a>
%H A038620 Thomas Bewley, Paul Belitz, and Joseph Cessna, <a href="http://robotics.ucsd.edu/pubs/BBC_Part1.pdf">New horizons in sphere packing theory, part I: fundamental concepts & constructions, from dense to rare</a>. See p. 18, row srs
%H A038620 J. K. Haugland, <a href="https://doi.org/10.1002/jgt.10071">Classification of certain subgraphs of the 3-dimensional grid</a>, J. Graph Theory, 42 (2003), 34-60.
%H A038620 J. K. Haugland, <a href="http://www.neutreeko.net/images/maths/gridgraph3.png">Illustration</a>
%H A038620 J. K. Haugland, <a href="/A038620/a038620.png">Illustration</a> [Cached copy, with permission] This illustration presents a different (less symmetric) embedding of the srs net into space.
%H A038620 M. O'Keeffe, <a href="https://doi.org/10.1524/zkri.1998.213.3.135">Coordination sequences for hyperbolic tilings</a>, Zeitschrift für Kristallographie, 213 (1998), 135-140 (see next-to-last table, row 10_5.10_5.10_5).
%H A038620 Reticular Chemistry Structure Resource, <a href="http://rcsr.net/nets/srs">srs</a>.
%H A038620 Toshikazu Sunada, <a href="https://www.ams.org/notices/200802/tx080200208p.pdf">Crystals that nature might miss creating</a>, Notices Amer. Math. Soc. 55 (No. 2, 2008), 208-215.
%H A038620 Toshikazu Sunada, <a href="https://www.ams.org/journals/notices/200803/tx080300342p.pdf#page=2">Correction to "Crystals That Nature Might Miss Creating"</a>, Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343.
%H A038620 Toshikazu Sunada, <a href="/A038620/a038620_1.png">Correction to "Crystals That Nature Might Miss Creating"</a>, Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343. [Annotated scanned copy]
%H A038620 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laves_graph">Laves graph</a>
%H A038620 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A038620 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.
%F A038620 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
%t A038620 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 *)
%t A038620 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 *)
%Y A038620 Cf. A038621 (partial sums), A290705 (theta series).
%K A038620 nonn,easy
%O A038620 0,2
%A A038620 _Jan Kristian Haugland_
%E A038620 Links corrected by _Jan Kristian Haugland_, Mar 01 2009
%E A038620 More terms from _Colin Barker_, May 10 2013