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 A215878 Lengths of loops in the P2 Penrose tiling. 0
 10, 20, 80, 100, 460, 620, 2780, 3700, 16660, 22220, 99980, 133300, 599860, 799820, 3599180, 4798900, 21595060, 28793420, 129570380, 172760500, 777422260, 1036563020, 4664533580, 6219378100, 27987201460, 37316268620, 167923208780, 223897611700, 1007539252660, 1343385670220, 6045235515980 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A loop of length n is defined to be an ordered set of n tiles (kites or darts), such that the tile T_i shares an edge with each of T_(i+1) and T_(i-1) (subscripts considered modulo n), but does not share a vertex with any other tile in the loop. These loops are the finite paths traced by gliders in a particular cellular automaton on the P2 Penrose tiling. LINKS Jacob Aron, First gliders navigate ever-changing Penrose universe, New Scientist. A. P. Goucher, Blog post about this A. P. Goucher, Gliders in Cellular Automata on Penrose Tilings, Journal of Cellular Automata (2012). Index entries for linear recurrences with constant coefficients, signature (0,5,0,6). FORMULA Recurrence relation: a(n+4) = 5*a(n+2) + 6*a(n). G.f.: -10*x*(3*x^2+2*x+1) / ((x^2+1)*(6*x^2-1)). - Colin Barker, May 19 2014 a(n) = 3*a(n-1)+2*a(n-2) if n is odd. a(n) = 2*a(n-1)-3*a(n-2) if n is even. - R. J. Mathar, Jun 18 2014 a(n) = -5 * ( -6^((n - 1)/2) * (9 + 2*sqrt(6) + (-1)^n * (2 * sqrt(6) - 9)) + 4 * (cos(n * Pi/2) + sin(n * Pi/2)))/7. - Benedict W. J. Irwin, Nov 01 2016 EXAMPLE The smallest loop a(1)=10 corresponds to the 10 kites which form the perimeter of a regular decagon. MATHEMATICA Table[{1, 1}.MatrixPower[{{5, 2}, {3, 0}}, Floor[n/2]].{10, 10Mod[n, 2]}, {n, 0, 49}] Table[-(5/7)(-6^(1/2(n-1))(9+2Sqrt[6]+(-1)^n(-9+2Sqrt[6]))+4(Cos[n Pi/2] + Sin[n Pi/2])), {n, 1, 20}] (* Benedict W. J. Irwin, Nov 01 2016 *) PROG (PARI) Vec(-10*x*(3*x^2+2*x+1)/((x^2+1)*(6*x^2-1)) + O(x^100)) \\ Colin Barker, May 19 2014 CROSSREFS Sequence in context: A267554 A131726 A276764 * A200985 A166641 A101244 Adjacent sequences:  A215875 A215876 A215877 * A215879 A215880 A215881 KEYWORD easy,nonn,walk AUTHOR Adam P. Goucher, Aug 25 2012 STATUS approved

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Last modified November 14 09:49 EST 2018. Contains 317182 sequences. (Running on oeis4.)