A341030 -id:A341030 - cp's OEIS frontend

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

Kevin Ryde, Feb 02 2021

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.

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

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).

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

Cf. A362566 (bounding box area).

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

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).

A343487 Decimal expansion of the perimeter of the convex hull around the terdragon fractal.

Original entry on oeis.org

2, 8, 1, 8, 8, 1, 4, 9, 2, 4, 8, 7, 0, 0, 6, 8, 8, 2, 0, 4, 6, 9, 7, 1, 6, 6, 8, 3, 1, 6, 1, 1, 2, 4, 6, 6, 3, 2, 4, 0, 3, 3, 0, 5, 3, 8, 2, 1, 8, 7, 2, 7, 1, 2, 6, 0, 9, 3, 1, 1, 1, 7, 4, 9, 1, 8, 6, 0, 2, 7, 5, 4, 4, 5, 9, 8, 4, 8, 5, 0, 5, 5, 4, 1, 7, 6, 5, 5, 3, 1, 5, 8, 0, 8, 4, 9, 5, 0, 1, 7, 1, 0, 3, 3, 3

Offset: 1

Kevin Ryde, Apr 17 2021

The convex hull around the terdragon fractal has 14 sides and with unit length from curve start to end their lengths are four sqrt(3)/24 and two each 1/24, 1/8, sqrt(3)/8, 3/8, sqrt(37)/12. Their total is the perimeter.

			2.8188149248700688204697166831611246...

Kevin Ryde, Table of n, a(n) for n = 1..10000
Kevin Ryde, Iterations of the Terdragon Curve, see index "HBf".

Cf. A343486 (terdragon hull area), A341030 (dragon hull perimeter).

RealDigits[(13+5*Sqrt[3]+2*Sqrt[37])/12,10,120][[1]] (* Harvey P. Dale, Dec 25 2021 *)

my(c=223+20*quadgen(3*37*4)); a_vector(len) = my(s=10^(len-1)); digits((13*s + sqrtint(floor(c*s^2))) \12);

Equals (13 + 5*sqrt(3) + 2*sqrt(37)) / 12.
Equals (13 + sqrt(223 + 20*sqrt(3*37))) / 12.
Largest root of ((12*x - 13)^2 - 223)^2 - 44400 = 0 (all its roots are real).

cp's OEIS Frontend

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

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