cp's OEIS Frontend

A298016 Coordination sequence of snub-632 tiling with respect to a hexavalent node.

%I A298016 #36 Apr 04 2020 01:39:14
%S A298016 1,6,12,12,24,36,24,42,60,36,60,84,48,78,108,60,96,132,72,114,156,84,
%T A298016 132,180,96,150,204,108,168,228,120,186,252,132,204,276,144,222,300,
%U A298016 156,240,324,168,258,348,180,276,372,192,294,396,204,312,420,216,330,444,228,348,468,240
%N A298016 Coordination sequence of snub-632 tiling with respect to a hexavalent node.
%C A298016 The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.
%C A298016 This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - _Felix Fröhlich_, Jan 13 2018
%D A298016 J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.
%H A298016 Colin Barker, <a href="/A298016/b298016.txt">Table of n, a(n) for n = 0..1000</a>
%H A298016 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 A298016 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="/A298016/a298016.png">Trunks and branches structure for calculating this sequence</a>
%H A298016 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H A298016 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 A298016 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A298016 For n >= 1, let k=floor(n/3). Then a(3*k) = 12*k, a(3*k+1)=18*k+6, a(3*k+2)=24*k+12.
%F A298016 a(n) = 2*a(n-3) - a(n-6) for n >= 7.
%F A298016 G.f.: -(-x^6-12*x^5-12*x^4-10*x^3-12*x^2-6*x-1)/(x^6-2*x^3+1).
%p A298016 f:=proc(n) local k,r;
%p A298016 if n=0 then return(1); fi;
%p A298016 r:=(n mod 3); k:=(n-r)/3;
%p A298016 if r=0 then 12*k elif r=1 then 18*k+6 else 24*k+12; fi;
%p A298016 end;
%p A298016 [seq(f(n),n=0..80)];
%t A298016 Join[{1}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {6, 12, 12, 24, 36, 24}, 60]] (* _Jean-François Alcover_, Apr 23 2018 *)
%o A298016 (PARI) Vec((1 + 6*x + 12*x^2 + 10*x^3 + 12*x^4 + 12*x^5 + x^6) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^60)) \\ _Colin Barker_, Jan 13 2018
%Y A298016 Cf. A298014, A298015.
%Y A298016 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 A298016 nonn,easy
%O A298016 0,2
%A A298016 Chaim Goodman-Strauss and _N. J. A. Sloane_, Jan 11 2018