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A038620
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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).
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8
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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A. F. Wells, Three-dimensional Nets and Polyhedra, Wiley, 1977. See the net (10,3)-a.
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LINKS
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J. K. Haugland, Illustration [Cached copy, with permission] This illustration presents a different (less symmetric) embedding of the srs net into space.
Reticular Chemistry Structure Resource, srs
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FORMULA
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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