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%I #32 Apr 04 2020 01:38:48
%S 1,3,12,9,24,15,36,21,48,27,60,33,72,39,84,45,96,51,108,57,120,63,132,
%T 69,144,75,156,81,168,87,180,93,192,99,204,105,216,111,228,117,240,
%U 123,252,129,264,135,276,141,288,147,300,153,312,159,324,165,336,171,348,177,360,183,372,189,384,195
%N Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.
%C Also known as the kgd net.
%C This is one of the Laves tilings.
%H Robert Israel, <a href="/A298028/b298028.txt">Table of n, a(n) for n = 0..10000</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/kgd">The kgd tiling (or net)</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.
%F G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - _Robert Israel_, Jan 21 2018
%F a(n) = 3*A022998(n), n>0. - _R. J. Mathar_, Jan 29 2018
%p f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
%p [seq(f3(n),n=0..80)];
%t Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* _Jean-François Alcover_, Mar 23 2020 *)
%Y Cf. A008579, A135556 (partial sums), A298026 (trivalent point).
%Y If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).
%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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