

A298682


Start with the triangle with 4 markings of the Shield tiling and recursively apply the substitution rule. a(n) is the number of triangles with 4 markings after n iterations.


6



1, 2, 4, 8, 28, 92, 352, 1280, 4828, 17900, 67024, 249680, 932716, 3479132, 12987904, 48464288, 180885628, 675045452, 2519361712, 9402270320, 35089981708, 130957132220, 488739595744, 1823999153600, 6807261212956, 25405037309612, 94812904802704
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OFFSET

0,2


COMMENTS

The following substitution rules apply to the tiles:
triangle with 6 markings > 1 hexagon
triangle with 4 markings > 1 square, 2 triangles with 4 markings
square > 1 square, 4 triangles with 6 markings
hexagon > 7 triangles with 6 markings, 3 triangles with 4 markings, 3 squares
a(n) is also one more than the number of squares after n iterations when starting with the triangle with 4 markings.


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
F. Gähler, Matching rules for quasicrystals: the compositiondecomposition method, Journal of NonCrystalline Solids, 153154 (1993), 160164.
Tilings Encyclopedia, Shield
Index entries for linear recurrences with constant coefficients, signature (3,5,9,2).


FORMULA

From Colin Barker, Jan 25 2018: (Start)
G.f.: (1 + x)*(1  2*x  5*x^2) / ((1  x)*(1 + 2*x)*(1  4*x + x^2)).
a(n) = (1/13)*(26 + (2)^n + (2+sqrt(3))^n*(7+5*sqrt(3))  (2sqrt(3))^n*(7+5*sqrt(3))).
a(n) = 3*a(n1) + 5*a(n2)  9*a(n3) + 2*a(n4) for n>3.
(End)


PROG

(PARI) /* The function substitute() takes as argument a 4element vector, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons that are to be substituted. The function returns a vector w, where the first, second, third and fourth elements respectively are the number of triangles with 6 markings, the number of triangles with 4 markings, the number of squares and the number of hexagons resulting from the substitution. */
substitute(v) = my(w=vector(4)); for(k=1, #v, while(v[1] > 0, w[4]++; v[1]); while(v[2] > 0, w[3]++; w[2]=w[2]+2; v[2]); while(v[3] > 0, w[3]++; w[1]=w[1]+4; v[3]); while(v[4] > 0, w[1]=w[1]+7; w[2]=w[2]+3; w[3]=w[3]+3; v[4])); w
terms(n) = my(v=[0, 1, 0, 0], i=0); while(1, print1(v[2], ", "); i++; if(i==n, break, v=substitute(v)))
(PARI) Vec((1 + x)*(1  2*x  5*x^2) / ((1  x)*(1 + 2*x)*(1  4*x + x^2)) + O(x^40)) \\ Colin Barker, Jan 25 2018


CROSSREFS

Cf. A298678, A298679, A298680, A298681, A298683.
Sequence in context: A259135 A219969 A034919 * A151340 A134316 A049461
Adjacent sequences: A298679 A298680 A298681 * A298683 A298684 A298685


KEYWORD

nonn,easy


AUTHOR

Felix Fröhlich, Jan 24 2018


EXTENSIONS

More terms from Colin Barker, Jan 25 2018


STATUS

approved



