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Coordination sequence of Dual(4.6.12) tiling with respect to a hexavalent node.
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%I #42 Jun 28 2024 23:06:20

%S 1,6,24,18,48,30,72,42,96,54,120,66,144,78,168,90,192,102,216,114,240,

%T 126,264,138,288,150,312,162,336,174,360,186,384,198,408,210,432,222,

%U 456,234,480,246,504,258,528,270,552,282,576,294,600,306,624,318,648

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

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

%C From _Keagan Boyce_, May 18 2024: (Start)

%C It appears that

%C a(n) = (3*n)*(3+(-1)^n) for n > 0,

%C which would imply that for all even n > 0,

%C a(n) = (3*n)*(3+(1)) = (3*n)*(4) = 12*n,

%C and for all odd n > 0,

%C a(n) = (3*n)*(3+(-1)) = (3*n)*(2) = 6*n. (End)

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

%H N. J. A. Sloane, <a href="/A298038/a298038.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].

%F Conjectures from _Colin Barker_, Apr 03 2020: (Start)

%F G.f.: (1 + 6*x + 22*x^2 + 6*x^3 + x^4) / ((1 - x)^2*(1 + x)^2).

%F a(n) = 2*a(n-2) - a(n-4) for n > 4. (End)

%F a(n) = (3*n)*(3+(-1)^n) for n > 0. - _Keagan Boyce_, May 18 2024

%t Table[(3*n)*(3+(-1)^(n)),{n,1,54}] (* _Keagan Boyce_, May 18 2024 *)

%Y Cf. A072154, A298039 (partial sums), A298036 (12-valent node), A298040 (tetravalent 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

%O 0,2

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

%E Terms a(8)-a(54) added by _Tom Karzes_, Apr 01 2020