A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.
1, 6, 12, 24, 30, 42, 48, 60, 66, 78, 84, 96, 102, 114, 120, 132, 138, 150, 156, 168, 174, 186, 192, 204, 210, 222, 228, 240, 246, 258, 264, 276, 282, 294, 300, 312, 318, 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
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
- Tom Karzes, Tiling Coordination Sequences
- Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
- N. J. A. Sloane, The Dual(3.4.6.4) tiling
- N. J. A. Sloane, The subgraph H shown in one 60-degree sector of the graph of the tiling.
- N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Crossrefs
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.
Programs
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Maple
f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end; [seq(f6(n),n=0..80)];
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Mathematica
Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* Jean-François Alcover, Apr 23 2018 *) -
PARI
Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018
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PARI
apply( {A298033(n)=if(n,n*3\/2*6-6,1)}, [0..66]) \\ M. F. Hasler, Jan 11 2022
Formula
Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)).
First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ...
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - Colin Barker, Jan 25 2018
a(n) = 6*floor((3n-1)/2) for n > 0. - M. F. Hasler, Jan 11 2022
Comments