%I #57 May 24 2020 23:12:29
%S 1,4,10,16,30,36,48,54,66,72,84,90,102,108,120,126,138,144,156,162,
%T 174,180,192,198,210,216,228,234,246,252,264,270,282,288,300,306,318,
%U 324,336,342,354,360,372,378,390,396,408,414,426,432,444,450,462,468,480,486,498,504,516,522,534,540
%N Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.
%C Also known as the mta net.
%C This is one of the Laves tilings.
%C In the Ferreol link this is described as the dual to the Diana tiling. - _N. J. A. Sloane_, May 24 2020
%H Colin Barker, <a href="/A298031/b298031.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="/A298031/a298031.png">The subgraph H shown in one quadrant of the graph of the 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F Theorem: For n >= 4, a(n) = 9*n-6 if n is even, otherwise a(n) = 9*n-9.
%F The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
%F G.f.: -(2*x^6 - 8*x^4 - 3*x^3 - 5*x^2 - 3*x - 1) / ((1 - x)*(1 - x^2)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>4. - _Colin Barker_, Jan 25 2018
%F a(n) = 6*A007494(n-1), n>3. - _R. J. Mathar_, Jan 29 2018
%p f4:=proc(n) local L; L:=[1,4,10,16];
%p if n<4 then L[n+1] elif (n mod 2) = 0 then 9*n-6 else 9*n-9; fi;
%p end;
%p [seq(f4(n),n=0..80)];
%t Join[{1, 4, 10, 16}, LinearRecurrence[{1, 1, -1}, {30, 36, 48}, 62]] (* _Jean-François Alcover_, Apr 23 2018 *)
%o (PARI) Vec((1 + 3*x + 5*x^2 + 3*x^3 + 8*x^4 - 2*x^6) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ _Colin Barker_, Jan 25 2018
%Y Cf. A008574, A298032 (partial sums), A298029 (for a trivalent 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; extended with formula, Jan 24 2018.