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

%I A343485 #12 Apr 25 2021 01:27:33
%S A343485 0,2,8,26,86,276,856,2586,7826,23628,71128,213546,641246,1925076,
%T A343485 5777416,17333706,52006586,156031788,468115048,1404358266,4213124006,
%U A343485 12639480276,37918617976,113755972026,341268358946,1023806051148,3071419747768,9214260306186
%N A343485 Area of the convex hull around terdragon expansion level n, measured in unit triangles.
%C A343485 Expansion level n comprises the first 3^n segments of the curve.
%H A343485 Kevin Ryde, <a href="/A343485/b343485.txt">Table of n, a(n) for n = 0..500</a>
%H A343485 Kevin Ryde, <a href="http://user42.tuxfamily.org/terdragon/index.html">Iterations of the Terdragon Curve</a>, see index "HA".
%H A343485 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,4,6,-36,36,-36,27).
%F A343485 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.
%F A343485 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.
%F A343485 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) ).
%F A343485 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) ).
%F A343485 Lim_{n->oo} a(n)/3^n = 29/24.
%e A343485 For n=1, the terdragon curve comprises 3 segments:
%e A343485     @---@      Convex hull vertices are marked "@".
%e A343485      \         They enclose an area of 2 unit triangles,
%e A343485   @---@        so a(1) = 2.
%e A343485 .
%e A343485 For n=2, the terdragon curve comprises 9 segments:
%e A343485     @---@
%e A343485      \         Convex hull vertices are marked "@".
%e A343485   @---*        They enclose an area of a(2) = 8
%e A343485    \ / \       unit triangle equivalents.
%e A343485     *---@
%e A343485      \
%e A343485   @---@
%o A343485 (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);
%Y A343485 Cf. A343486 (fractal hull area), A341029 (dragon curve hull area).
%K A343485 nonn,easy
%O A343485 0,2
%A A343485 _Kevin Ryde_, Apr 17 2021