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 A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node. 22
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also known as the mta net. This is one of the Laves tilings. 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, 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). 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 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)]; MATHEMATICA Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* Jean-François Alcover, Apr 23 2018 *) PROG (PARI) Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018 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. Cf. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node). Sequence in context: A307225 A261476 A119840 * A069171 A071611 A216453 Adjacent sequences:  A298030 A298031 A298032 * A298034 A298035 A298036 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Jan 21 2018, corrected Jan 24 2018. STATUS approved

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Last modified September 24 07:28 EDT 2020. Contains 337317 sequences. (Running on oeis4.)