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%I #47 Jan 24 2022 17:12:06
%S 1,3,21,39,57,75,93,111,129,147,165,183,201,219,237,255,273,291,309,
%T 327,345,363,381,399,417,435,453,471,489,507,525,543,561,579,597,615,
%U 633,651,669,687,705,723,741,759,777,795,813,831,849,867,885,903,921,939,957,975,993,1011,1029,1047,1065
%N Coordination sequence of Dual(3.12.12) tiling with respect to a trivalent node.
%C This tiling is sometimes called the triakis triangular tiling.
%H Colin Barker, <a href="/A298035/b298035.txt">Table of n, a(n) for n = 0..1000</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="https://arxiv.org/abs/1803.08530">on arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.
%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H N. J. A. Sloane, <a href="/A298035/a298035.png">Illustration of initial terms</a> (shows one 120-degree sector of graph).
%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_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F Theorem: a(0)=1; thereafter a(n) = 18*n-15. [Proof: Use the "coloring book" method described in the Goodman-Strauss & Sloane article.]
%F From _Colin Barker_, Jan 22 2018: (Start)
%F G.f.: (1 + x + 16*x^2) / (1 - x)^2.
%F a(n) = 2*a(n-1) - a(n-2) for n>2.
%F (End)
%p f3:=proc(n) if n=0 then 1 else 18*n-15; fi; end;
%p [seq(f3(n),n=0..80)];
%o (PARI) Vec((1 + x + 16*x^2) / (1 - x)^2 + O(x^60)) \\ _Colin Barker_, Jan 22 2018
%Y Cf. A019557 (12-valent node), A016790 (partial sums, provided its offset is changed).
%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 22 2018
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