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 A298681 Start with the square tile of the Shield tiling and recursively apply the substitution rule. a(n) is the number of triangles with 6 markings after n iterations. 6
 0, 4, 4, 32, 80, 372, 1236, 4912, 17728, 67364, 248996, 934080, 3476400, 12993364, 48453364, 180907472, 675001760, 2519449092, 9402095556, 35090331232, 130956433168, 488740993844, 1823996357396, 6807266805360, 25405026124800, 94812927172324, 353846503607524 (list; graph; refs; listen; history; text; internal format)
 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 > 0, w++; v--); while(v > 0, w++; w=w+2; v--); while(v > 0, w++; w=w+4; v--); while(v > 0, w=w+7; w=w+3; w=w+3; v--)); w terms(n) = my(v=[0, 0, 1, 0], i=0); while(1, print1(v, ", "); 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 Cf. A298678, A298679, A298680, A298682, A298683. Sequence in context: A271019 A271003 A217310 * A189743 A089542 A222285 Adjacent sequences:  A298678 A298679 A298680 * A298682 A298683 A298684 KEYWORD nonn,easy AUTHOR Felix Fröhlich, Jan 24 2018 EXTENSIONS More terms from Colin Barker, Jan 25 2018 STATUS approved

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Last modified October 18 21:24 EDT 2021. Contains 348070 sequences. (Running on oeis4.)