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

0,2

Comments

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

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).

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.

Example

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.
    *---@
     \
  @---@

Prog

(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);

Crossrefs

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

Sequence in context: A128634 A230904 A289595 * A298189 A053956 A320802

Adjacent sequences: A343482 A343483 A343484 * A343486 A343487 A343488

Keyword

nonn,easy

Author

Kevin Ryde, Apr 17 2021

Status

approved