A341029 Twice the area of the convex hull around dragon curve expansion level n.

0, 1, 3, 9, 23, 56, 121, 258, 539, 1118, 2273, 4614, 9323, 18806, 37761, 75798, 151979, 304598, 609793, 1220694, 2442923, 4888406, 9779201, 19562838, 39131819, 78273878, 156557313, 313132374, 626289323, 1252619606, 2505277441, 5010625878, 10021350059

Offset: 0

Author

Kevin Ryde, Feb 02 2021

Keywords

• nonn
• easy

Comments

The area of the hull is a half-integer for n=1..4 and even n>=6, so the sequence is a(n) = 2*area to give integers.

Benedek and Panzone determine the vertices of the convex hull around the dragon fractal. The area of that hull is 7/6 (A177057). This is the limit for the finite expansions scaled down to a unit distance start to end: lim_{n->oo} (a(n)/2) / 2^n = 7/6.

Examples

			  @   *---@          curve expansion level n=3,
  |   |   |          convex hull vertices marked "@",
  @---*   *---@      area = 4+1/2,
              |      a(3) = 2*area = 9
          @---@
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..600
• Agnes I. Benedek and Rafael Panzone, On Some Notable Plane Sets, II: Dragons, Revista de la Unión Matemática Argentina, volume 39, numbers 1-2, 1994, pages 76-90.
• Kevin Ryde, Iterations of the Dragon Curve, see index "HA".
• Index entries for linear recurrences with constant coefficients, signature (3,-3,3,2,-12,12,-12,8).

Crossrefs

Cf. A177057 (fractal hull area), A341030 (fractal hull perimeter).

Cf. A362566 (bounding box area).

Programs

PARI

my(h=[22,29,22,31]); a(n) = if(n<2,n, (7<
				

Formula

For n>=2, a(n) = (7/3)*2^n - (h/6)*2^floor(n/2) + c/3, where h = 22,29,22,31 and c = 1,2,3,2 according as n == 0,1,2,3 (mod 4) respectively.

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) + 2*a(n-4) - 12*a(n-5) + 12*a(n-6) - 12*a(n-7) + 8*a(n-8) for n>=10.

G.f.: x*(1 + 3*x^2 + 2*x^3 + 3*x^4 + x^5 - 2*x^7 - 4*x^8) /( (1-x) * (1-2*x) * (1+x^2) * (1-2*x^2) * (1+2*x^2) ).

G.f.: 1 + (1/2)*x + (2/3)/(1-x) - (1/3)/(1+x^2) + (1/6)*x/(1+2*x^2) - (11/3 + 5*x)/(1-2*x^2) + (7/3)/(1-2*x).