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

0,3

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.

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).

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).

Example

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

Prog

(PARI) my(h=[22, 29, 22, 31]); a(n) = if(n<2, n, (7<<n - h[n%4+1]<<(n\2-1))\3 + 1);

Crossrefs

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

Cf. A362566 (bounding box area).

Sequence in context: A274099 A147126 A147212 * A045883 A292329 A133654

Adjacent sequences: A341026 A341027 A341028 * A341030 A341031 A341032

Keyword

nonn,easy

Author

Kevin Ryde, Feb 02 2021

Status

approved