A343485 Area of the convex hull around terdragon expansion level n, measured in unit triangles.

0, 2, 8, 26, 86, 276, 856, 2586, 7826, 23628, 71128, 213546, 641246, 1925076, 5777416, 17333706, 52006586, 156031788, 468115048, 1404358266, 4213124006, 12639480276, 37918617976, 113755972026, 341268358946, 1023806051148, 3071419747768, 9214260306186

Offset: 0

Author

Kevin Ryde, Apr 17 2021

Keywords

• nonn
• easy

Comments

Expansion level n comprises the first 3^n segments of the curve.

Examples

			For n=1, the terdragon curve comprises 3 segments:
    @---@      Convex hull vertices are marked "@".
     \         They enclose an area of 2 unit triangles,
  @---@        so a(1) = 2.
.
For n=2, the terdragon curve comprises 9 segments:
    @---@
     \         Convex hull vertices are marked "@".
  @---*        They enclose an area of a(2) = 8
   \ / \       unit triangle equivalents.
    *---@
     \
  @---@
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..500
• Kevin Ryde, Iterations of the Terdragon Curve, see index "HA".
• Index entries for linear recurrences with constant coefficients, signature (4,-4,4,6,-36,36,-36,27).

Crossrefs

Cf. A343486 (fractal hull area), A341029 (dragon curve hull area).

Programs

PARI

my(h=[30,46,22,50]); a(n) = if(n<2,2*n, (29*3^n - h[n%4+1]*3^(n\2))\24);

Formula

For n>=2, a(n) = (29/24)*3^n - (h/12)*3^floor(n/2) - (c/8) where h = 15,23,11,25 and c = 5,3,1,3 according as n == 0,1,2,3 (mod 4) respectively.

a(n) = 4*a(n-1) - 4*a(n-2) + 4*a(n-3) + 6*a(n-4) - 36*a(n-5) + 36*a(n-6) - 36*a(n-7) + 27*a(n-8), for n>=10.

G.f.: (2*x + 2*x^3 + 6*x^4 - 8*x^5 + 16*x^6 - 18*x^7 + 6*x^8 - 18*x^9) /( (1-x)*(1+x^2)*(1-9*x^4)*(1-3*x) ).

G.f.: (1/24)*( 16 + 16*x - 9/(1-x) - 6/(1+x^2) - (26+48*x)/(1-3*x^2) + (-4+2*x)/(1+3*x^2) + 29/(1-3*x) ).

Lim_{n->oo} a(n)/3^n = 29/24.