A215878 Lengths of loops in the P2 Penrose tiling.

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

Offset: 1

Author

Adam P. Goucher, Aug 25 2012

Keywords

• easy
• nonn
• walk

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.

Examples

			The smallest loop a(1)=10 corresponds to the 10 kites which form the perimeter of a regular decagon.
		

Links

• Jacob Aron, First gliders navigate ever-changing Penrose universe, New Scientist.
• Adam P. Goucher, Blog post about this
• Adam 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).

Programs

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 *)

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

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