A227036 Expansion of 2*(1+x^2)/((1-x)*(1-x-2*x^3)).

2, 4, 8, 16, 28, 48, 84, 144, 244, 416, 708, 1200, 2036, 3456, 5860, 9936, 16852, 28576, 48452, 82160, 139316, 236224, 400548, 679184, 1151636, 1952736, 3311108, 5614384, 9519860, 16142080, 27370852, 46410576, 78694740, 133436448, 226257604, 383647088, 650519988, 1103035200, 1870329380

Offset: 0

Author

Roland Kneer, Jun 28 2013

Keywords

• nonn
• easy

Comments

Conjecture: The perimeter of the n-th iteration of the Harter-Heighway dragon is a(n) segments or a(n)/2^(n/2) base units.

a(n) = 2^(n+1)-4*A003230(n-4) (two times the number of segments, minus four times the number of squares)

The first differences 2, 2, 4, 8, 12, 20,.. are twice the (empirical) A203175. - R. J. Mathar, Jul 02 2013

Examples

			For the 4th iteration, take two 3rd iteration dragons (2*16); put together, they will make one square, so subtract the inner perimeter 4.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kevin Ryde, Iterations of the Dragon Curve, see index "B" and "R".
• Helena Verrill, On the Boundary of the Harter-Heighway dragon curve, arXiv:2407.17326 [math.CO], 2024.
• Wikipedia, Dragon curve
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-2).

Crossrefs

Cf. A014577.

Programs

Mathematica

LinearRecurrence[{2, -1, 2, -2}, {2, 4, 8, 16}, 40] (* T. D. Noe, Jul 02 2013 *)
CoefficientList[Series[2 (1 + x^2) / ((1 - x) (1 - x - 2 x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 17 2013 *)

PARI

Vec(2*(1+x^2)/((1-x)*(1-x-2*x^3))+O(x^66)) \\ Joerg Arndt, Jul 01 2013