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A298015 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-short. 22
1, 3, 6, 15, 24, 18, 33, 48, 30, 51, 72, 42, 69, 96, 54, 87, 120, 66, 105, 144, 78, 123, 168, 90, 141, 192, 102, 159, 216, 114, 177, 240, 126, 195, 264, 138, 213, 288, 150, 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 (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 C. 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

C. Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, arXiv:1803.08530 [math.CO], March 2018.

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 >= 5, let k=floor(n/3). Then a(3*k) = 18*k-3, a(3*k+1)=24*k, a(3*k+2)=12*k+6.

a(n) = 2*a(n-3) - a(n-6) for n>=8.

G.f.: -(-3*x^7-4*x^6-6*x^5-18*x^4-13*x^3-6*x^2-3*x-1)/(x^6-2*x^3+1).

MAPLE

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

if n=0 then return(1) elif n=1 then return(3); fi;

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

if r=0 then 18*k-3 elif r=1 then 24*k else 12*k+6 fi;

end;

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

MATHEMATICA

Join[{1, 3, 6, 15, 24}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {18, 33, 48, 30, 51, 72}, 59]] (* Jean-François Alcover, Apr 28 2018 *)

PROG

(PARI) Vec((1 + 3*x + 6*x^2 + 13*x^3 + 18*x^4 + 6*x^5 + 4*x^6 + 3*x^7) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jan 13 2018

CROSSREFS

Cf. A298014, A298016.

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: A180322 A244164 A129602 * A044888 A179805 A006639

Adjacent sequences:  A298012 A298013 A298014 * A298016 A298017 A298018

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified January 22 12:13 EST 2019. Contains 319363 sequences. (Running on oeis4.)