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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

Table of n, a(n) for n=1..31.

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 June 26 13:33 EDT 2017. Contains 288766 sequences.