|
%I #67 Jan 11 2022 21:57:16
%S 1,6,12,24,30,42,48,60,66,78,84,96,102,114,120,132,138,150,156,168,
%T 174,186,192,204,210,222,228,240,246,258,264,276,282,294,300,312,318,
%U 330,336,348,354,366,372,384,390,402,408,420,426,438,444,456,462,474,480,492,498,510,516,528,534,546,552
%N Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.
%C Also known as the mta net.
%C This is one of the Laves tilings.
%H Colin Barker, <a href="/A298033/b298033.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="http://arxiv.org/abs/1803.08530">arXiv:1803.08530</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/mta">The mta tiling (or net)</a>
%H N. J. A. Sloane, <a href="/A298029/a298029.png">The Dual(3.4.6.4) tiling</a>
%H N. J. A. Sloane, <a href="/A298033/a298033.png">The subgraph H shown in one 60-degree sector of the graph of the tiling.</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_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd.
%F The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
%F G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)).
%F First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ...
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - _Colin Barker_, Jan 25 2018
%F a(n) = 6*floor((3n-1)/2) for n > 0. - _M. F. Hasler_, Jan 11 2022
%p f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end;
%p [seq(f6(n),n=0..80)];
%t Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* _Jean-François Alcover_, Apr 23 2018 *)
%o (PARI) Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ _Colin Barker_, Jan 25 2018
%o (PARI) apply( {A298033(n)=if(n,n*3\/2*6-6,1)}, [0..66]) \\ _M. F. Hasler_, Jan 11 2022
%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.
%Y Cf. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 21 2018, corrected Jan 24 2018.
|
