OFFSET
0,1
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
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).
EXAMPLE
For the 4th iteration, take two 3rd iteration dragons (2*16); put together, they will make one square, so subtract the inner perimeter 4.
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 *)
PROG
(PARI) Vec(2*(1+x^2)/((1-x)*(1-x-2*x^3))+O(x^66)) \\ Joerg Arndt, Jul 01 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roland Kneer, Jun 28 2013
STATUS
approved