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%I #37 Jan 19 2020 18:13:17
%S 1,3,9,15,18,27,37,37,44,57,54,61,77,71,78,97,88,95,117,105,112,137,
%T 122,129,157,139,146,177,156,163,197,173,180,217,190,197,237,207,214,
%U 257,224,231,277,241,248,297,258,265,317,275,282,337,292,299,357,309,316,377,326,333,397,343,350,417,360
%N Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-long.
%C 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 This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - _Felix Fröhlich_, Jan 13 2018
%D 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 Colin Barker, <a href="/A298014/b298014.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 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_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F For n >= 5, let k=floor(n/3). Then a(3*k) = 20*k-3, a(3*k+1)=17*k+3, a(3*k+2)=17*k+10.
%F G.f.: -(x+1)*(2*x^9+x^7-5*x^6-3*x^5-6*x^4-6*x^3-7*x^2-2*x-1)/((x-1)^2*(x^2+x+1)^2).
%F a(n) = 2*a(n-3) - a(n-6) for n>10. - _Colin Barker_, Jan 13 2018
%p f:=proc(n) local k,r,L; L:=[1,3,9,15,18];
%p if n<5 then L[n+1]
%p else k:=floor(n/3); r:=n-3*k;
%p if r=0 then 20*k-3 elif r=1 then 17*k+3 else 17*k+10; fi;
%p fi; end;
%p [seq(f(n),n=0..80)];
%t Join[{1, 3, 9, 15, 18}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {27, 37, 37, 44, 57, 54}, 60]] (* _Jean-François Alcover_, Apr 28 2018 *)
%o (PARI) Vec(-(x+1)*(2*x^9+x^7-5*x^6-3*x^5-6*x^4-6*x^3-7*x^2-2*x-1)/((x-1)^2*(x^2+x+1)^2) + O(x^60)) \\ _Colin Barker_, Jan 13 2018
%Y Cf. A298015, A298016.
%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,easy
%O 0,2
%A Chaim Goodman-Strauss and _N. J. A. Sloane_, Jan 11 2018
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