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A215878
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Lengths of loops in the P2 Penrose tiling.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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The smallest loop a(1)=10 corresponds to the 10 kites which form the perimeter of a regular decagon.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn,walk
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AUTHOR
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STATUS
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approved
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