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A298016 Coordination sequence of snub-632 tiling with respect to a hexavalent node. 22
1, 6, 12, 12, 24, 36, 24, 42, 60, 36, 60, 84, 48, 78, 108, 60, 96, 132, 72, 114, 156, 84, 132, 180, 96, 150, 204, 108, 168, 228, 120, 186, 252, 132, 204, 276, 144, 222, 300, 156, 240, 324, 168, 258, 348, 180, 276, 372, 192, 294, 396, 204, 312, 420, 216, 330, 444, 228, 348, 468, 240 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.

This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - Felix Fröhlich, Jan 13 2018

REFERENCES

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.

Chaim Goodman-Strauss and N. J. A. Sloane, Trunks and branches structure for calculating this sequence

Tom Karzes, Tiling Coordination Sequences

N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

FORMULA

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.

a(n) = 2*a(n-3) - a(n-6) for n >= 7.

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).

MAPLE

f:=proc(n) local k, r;

if n=0 then return(1); fi;

r:=(n mod 3); k:=(n-r)/3;

if r=0 then 12*k elif r=1 then 18*k+6 else 24*k+12; fi;

end;

[seq(f(n), n=0..80)];

MATHEMATICA

Join[{1}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {6, 12, 12, 24, 36, 24}, 60]] (* Jean-François Alcover, Apr 23 2018 *)

PROG

(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

CROSSREFS

Cf. A298014, A298015.

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.

Sequence in context: A330922 A330921 A183688 * A055595 A132632 A223352

Adjacent sequences:  A298013 A298014 A298015 * A298017 A298018 A298019

KEYWORD

nonn,easy

AUTHOR

Chaim Goodman-Strauss and N. J. A. Sloane, Jan 11 2018

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)