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%I A296910 #42 Aug 30 2023 17:05:53
%S A296910 1,4,6,14,14,22,22,30,30,38,38,46,46,54,54,62,62,70,70,78,78,86,86,94,
%T A296910 94,102,102,110,110,118,118,126,126,134,134,142,142,150,150,158,158,
%U A296910 166,166,174,174,182,182,190,190,198,198,206,206,214,214,222,222,230,230,238,238
%N A296910 a(0)=1, a(1)=4; thereafter a(n) = 4*n-2*(-1)^n.
%C A296910 Coordination sequence for the bew tiling with respect to a point where two hexagons meet at only a single point. The coordination sequence for the other type of point can be shown to be A008574.
%C A296910 Notes: There is one point on the positive x-axis at edge-distance n from the origin iff n is even; there is one point on the positive y-axis at edge-distance n from the origin iff n>1 is odd; and the number of points inside the first quadrant at distance n from 0 is n if n is odd, and n-1 if n is even.
%C A296910 Then a(n) = 2*(number on positive x-axis + number on positive y-axis) + 4*(number in interior of first quadrant).
%H A296910 Colin Barker, <a href="/A296910/b296910.txt">Table of n, a(n) for n = 0..1000</a>
%H A296910 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A296910 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 A296910 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/bew">The bew tiling (or net)</a>
%H A296910 Anton Shutov and Andrey Maleev, <a href="https://doi.org/10.1515/zkri-2020-0002">Coordination sequences of 2-uniform graphs</a>, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
%H A296910 N. J. A. Sloane, <a href="/A296910/a296910.jpg">Illustration of initial terms.</a> Points in the first quadrant are marked with their edge-distance from the origin (the heavy black circle). (Ignore the black rectangles, which show some fundamental cells for this tiling.)
%H A296910 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A296910 From _Colin Barker_, Dec 23 2017: (Start)
%F A296910 G.f.: (1 + 3*x + x^2 + 5*x^3 - 2*x^4) / ((1 - x)^2*(1 + x)).
%F A296910 a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
%F A296910 (End)
%t A296910 {1, 4}~Join~Array[4 # - 2 (-1)^# &, 59, 2] (* or *)
%t A296910 LinearRecurrence[{1, 1, -1}, {1, 4, 6, 14, 14}, 61] (* or *)
%t A296910 CoefficientList[Series[(1 + 3 x + x^2 + 5 x^3 - 2 x^4)/((1 - x)^2*(1 + x)), {x, 0, 60}], x] (* _Michael De Vlieger_, Dec 23 2017 *)
%o A296910 (PARI) Vec((1 + 3*x + x^2 + 5*x^3 - 2*x^4) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ _Colin Barker_, Dec 23 2017
%Y A296910 Apart from first two terms, same as A168384.
%Y A296910 Cf. A008574. See A296911 for partial sums.
%Y A296910 Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
%K A296910 nonn,easy
%O A296910 0,2
%A A296910 _N. J. A. Sloane_, Dec 22 2017

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