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Coordination sequence of Dual(4.6.12) tiling with respect to a 12-valent node.
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%I #42 Jul 19 2024 19:06:29

%S 1,12,12,36,24,60,36,84,48,108,60,132,72,156,84,180,96,204,108,228,

%T 120,252,132,276,144,300,156,324,168,348,180,372,192,396,204,420,216,

%U 444,228,468,240,492,252,516,264,540,276,564,288,588,300

%N Coordination sequence of Dual(4.6.12) tiling with respect to a 12-valent node.

%C Conjecture: For n>0, a(n)=6n if n even, otherwise 12n.

%C The conjecture can easily be shown to be true: The vertices at distance 2k consist of 3k 12-valent and 3k 4-alent vertices, and the vertices at distance 2k+1 consist of 6(k+1) 6-valent and 6(k+1) 4-valent vertices. - _Charlie Neder_, Apr 22 2019

%H Hakan Icoz, <a href="/A298036/b298036.txt">Table of n, a(n) for n = 0..20000</a>

%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>

%H N. J. A. Sloane, <a href="/A298036/a298036.png">Illustration of initial terms</a> (shows one 60-degree sector of tiling)

%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 From _Charlie Neder_, Apr 22 2019: (Start)

%F a(n) = 6 * n * (1 + n mod 2), n > 0.

%F G.f.: (1 + 12*x + 10*x^2 + 12*x^3 + x^4)/(1 - x^2)^2. (End)

%t LinearRecurrence[{0, 2, 0, -1}, {1, 12, 12, 36, 24}, 100] (* _Paolo Xausa_, Jul 19 2024 *)

%Y Cf. A072154, A298037 (partial sums), A298038 (hexavalent node), A298040 (tetravalent node).

%Y Cf. A109043 (a(n)/6), A026741 (a(n)/12).

%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

%O 0,2

%A _N. J. A. Sloane_, Jan 22 2018

%E a(7)-a(50) from _Charlie Neder_, Apr 22 2019