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%I #27 Mar 23 2020 06:39:17
%S 1,6,6,18,12,30,18,42,24,54,30,66,36,78,42,90,48,102,54,114,60,126,66,
%T 138,72,150,78,162,84,174,90,186,96,198,102,210,108,222,114,234,120,
%U 246,126,258,132,270,138,282,144,294,150,306,156,318,162,330,168,342,174,354,180,366,186,378,192,390
%N Coordination sequence of Dual(3.6.3.6) tiling with respect to a hexavalent node.
%C Also known as the kgd net.
%C This is one of the Laves tilings.
%H Robert Israel, <a href="/A298026/b298026.txt">Table of n, a(n) for n = 0..10000</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)=6*k, a(2*k+1)=12*k+6.
%F G.f.: 1 + 6*x*(1+x+x^2)/(1-x^2)^2. - _Robert Israel_, Jan 21 2018
%F From _Colin Barker_, Jan 22 2018: (Start)
%F a(n) = 3*n for n>0 and even.
%F a(n) = 6*n for n odd.
%F a(n) = 2*a(n-2) - a(n-4) for n>4.
%F (End)
%F a(n) = 6*A026741(n), n>0. - _R. J. Mathar_, Jan 29 2018
%p f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 3*n else 6*n; fi; end;
%p [seq(f6(n),n=0..80)];
%t Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {6, 6, 18, 12}, 80]] (* _Jean-François Alcover_, Mar 23 2020 *)
%o (PARI) Vec((1 + 6*x + 4*x^2 + 6*x^3 + x^4) / ((1 - x)^2*(1 + x)^2) + O(x^60)) \\ _Colin Barker_, Jan 22 2018
%Y Cf. A008579, A298027 (partial sums), A298028 (trivalent point).
%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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