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%I #62 May 24 2020 23:14:29
%S 1,3,6,12,18,33,39,51,57,69,75,87,93,105,111,123,129,141,147,159,165,
%T 177,183,195,201,213,219,231,237,249,255,267,273,285,291,303,309,321,
%U 327,339,345,357,363,375,381,393,399,411,417,429,435,447,453,465,471,483,489,501,507,519,525,537,543,555
%N Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.
%C Also known as the deltoidal trihexagonal tiling, or the mta net.
%C In the Ferreol link this is described as the dual to the Diana tiling. - _N. J. A. Sloane_, May 24 2020
%C This is one of the Laves tilings.
%H Colin Barker, <a href="/A298029/b298029.txt">Table of n, a(n) for n = 0..1000</a>
%H Robert Ferreol, <a href="https://mathcurve.com/polyedres/pavagedediane/pavagedediane.shtml">Pavage de diane et son dual</a>
%H Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">arXiv:1803.08530</a>.
%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/mta">The mta tiling (or net)</a>
%H N. J. A. Sloane, <a href="/A298029/a298029.png">The Dual(3.4.6.4) tiling</a>
%H N. J. A. Sloane, <a href="/A296368/a296368_2.png">Overview of coordination sequences of Laves tilings</a> [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_convex_uniform_tilings#Laves_tilings">Laves tilings</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F Theorem: For n >= 5, if n is even then a(n) = 9*n-15, otherwise a(n) = 9*n-12. The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. - _N. J. A. Sloane_, Jan 24 2018
%F G.f.: -(3*x^7 - 9*x^5 - 3*x^4 - 4*x^3 - 2*x^2 - 2*x - 1)/((1 - x)*(1 - x^2)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>5. - _Colin Barker_, Jan 25 2018
%F a(n) = (3/2)*(6*n - (-1)^n - 9) for n>4. - _Bruno Berselli_, Jan 25 2018
%F a(n) = 3*A007310(n-1), n>4. - _R. J. Mathar_, Jan 29 2018
%t Join[{1, 3, 6, 12, 18}, LinearRecurrence[{1, 1, -1}, {33, 39, 51}, 60]] (* _Jean-François Alcover_, Jan 07 2019 *)
%t Join[{1,3,6,12,18},Table[If[EvenQ[n],9n-15,9n-12],{n,5,70}]] (* _Harvey P. Dale_, Aug 25 2019 *)
%o (PARI) Vec((1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 9*x^5 - 3*x^7) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ _Colin Barker_, Jan 25 2018
%Y Cf. A008574, A298030 (partial sums), A298031 (for a tetravalent node), A298033 (hexavalent node).
%Y List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 21 2018
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