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
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
F. Gähler, Matching rules for quasicrystals: the composition-decomposition method, Journal of Non-Crystalline Solids, 153-154 (1993), 160-164.
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.: 4*x*(1 - 2*x) / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)).
a(n) = (1/39)*(26 + (-1)^(1+n)*2^(5+n) + (3-9*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(3+9*sqrt(3))).
a(n) = 3*a(n-1) + 5*a(n-2) - 9*a(n-3) + 2*a(n-4) for n>3.
(End)
PROG
(PARI) /* The function substitute() takes as argument a 4-element 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, 0, 1, 0], i=0); while(1, print1(v[1], ", "); i++; if(i==n, break, v=substitute(v)))
(PARI) concat(0, Vec(4*x*(1 - 2*x) / ((1 - x)*(1 + 2*x)*(1 - 4*x + x^2)) + O(x^40))) \\ Colin Barker, Jan 25 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Felix Fröhlich, Jan 24 2018
EXTENSIONS
More terms from Colin Barker, Jan 25 2018
STATUS
approved