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A298022 Coordination sequence for Dual(3^3.4^2) tiling with respect to a trivalent node.

%I A298022 #47 Nov 28 2024 11:08:07
%S A298022 1,3,7,12,17,23,28,33,37,42,47,51,56,61,65,70,75,79,84,89,93,98,103,
%T A298022 107,112,117,121,126,131,135,140,145,149,154,159,163,168,173,177,182,
%U A298022 187,191,196,201,205,210,215,219,224,229,233,238,243,247,252,257,261
%N A298022 Coordination sequence for Dual(3^3.4^2) tiling with respect to a trivalent node.
%C A298022 This tiling is also called the prismatic pentagonal tiling, or the cem-d net. It is one of the 11 Laves tilings.
%D A298022 B. Gruenbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. See p. 96.
%H A298022 Rémy Sigrist, <a href="/A298022/b298022.txt">Table of n, a(n) for n = 0..1000</a>
%H A298022 Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. <a href="http://dx.doi.org/10.1090/noti838">Isoperimetric Pentagonal Tilings</a>, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (right).
%H A298022 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H A298022 Frank Morgan, <a href="https://www.youtube.com/watch?v=PpUx0nnWfKQ">Optimal Pentagonal Tilings</a>, Video, May 2021. [Mentions this tiling]
%H A298022 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/cem-d">The cem-d tiling (or net)</a>
%H A298022 Rémy Sigrist, <a href="/A298022/a298022.gp.txt">PARI program for A298022</a>
%H A298022 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 A298022 N. J. A. Sloane, <a href="/A298022/a298022.png">Illustration of initial terms</a>
%F A298022 Conjectures from _Colin Barker_, Jan 22 2018: (Start)
%F A298022 G.f.: (1 + 2*x + 4*x^2 + 4*x^3 + 3*x^4 + 2*x^5 - 2*x^8) / ((1 - x)^2*(1 + x + x^2)).
%F A298022 a(n) = a(n-1) + a(n-3) - a(n-4) for n>5. (End)
%F A298022 Conjecture: a(n) = 2*(21*n + 3*A049347(n+2)/2)/9 for n > 4. - _Stefano Spezia_, Nov 24 2024
%o A298022 (PARI) \\ See Links section.
%Y A298022 See A298023 for partial sums, A298024 for a tetravalent point.
%Y A298022 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.
%Y A298022 Cf. A049347.
%K A298022 nonn
%O A298022 0,2
%A A298022 _N. J. A. Sloane_, Jan 21 2018
%E A298022 More terms from _Rémy Sigrist_, Jan 21 2018