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A298015 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-short. 22

%I #42 Jan 18 2021 17:06:23

%S 1,3,6,15,21,18,33,48,30,51,72,42,69,96,54,87,120,66,105,144,78,123,

%T 168,90,141,192,102,159,216,114,177,240,126,195,264,138,213,288,150,

%U 231,312,162,249,336,174,267,360,186,285,384,198,303,408,210,321,432,222,339,456,234,357,480,246,375

%N Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-short.

%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="/A298015/b298015.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>. [Warning: there is an error in Eq. 8(b), the a(4) term should be changed from 24 to 21. With that correction Theorem then still holds. - _N. J. A. Sloane_, Apr 01 2020]

%H Tom Karzes, <a href="/A298015/a298015.gif">Illustration of a(0) to a(4)</a> [Key: n, a(n), color: 0, 1, green; 1, 3, red; 2, 6, blue; 3, 15, purple; 4, 21, beige.]

%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 >= 6, let k=floor(n/3), so k >= 2. Then a(3*k) = 18*k-3, a(3*k+1)=24*k, a(3*k+2)=12*k+6. [Corrected by _N. J. A. Sloane_, Apr 01 2020]

%F a(n) = 2*a(n-3) - a(n-6) for n>=11. [Corrected by _N. J. A. Sloane_, Apr 01 2020]

%F G.f.: -(3*x^10-9*x^7-4*x^6-6*x^5-15*x^4-13*x^3-6*x^2-3*x-1)/(x^6-2*x^3+1). [Corrected by _N. J. A. Sloane_, Apr 01 2020]

%Y Cf. A298014, 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

%E a(4) corrected by _Tom Karzes_. I corrected the b-file and the formulas and deleted the programs. - _N. J. A. Sloane_, Apr 01 2020

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