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 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 (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 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 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.: (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)) - (2-sqrt(3))^n*(7+5*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, 1, 0, 0], i=0); while(1, print1(v, ", "); 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

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Last modified October 6 04:36 EDT 2022. Contains 357261 sequences. (Running on oeis4.)