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

%I A341029 #17 Apr 28 2023 08:15:55
%S A341029 0,1,3,9,23,56,121,258,539,1118,2273,4614,9323,18806,37761,75798,
%T A341029 151979,304598,609793,1220694,2442923,4888406,9779201,19562838,
%U A341029 39131819,78273878,156557313,313132374,626289323,1252619606,2505277441,5010625878,10021350059
%N A341029 Twice the area of the convex hull around dragon curve expansion level n.
%C A341029 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.
%C A341029 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.
%H A341029 Kevin Ryde, <a href="/A341029/b341029.txt">Table of n, a(n) for n = 0..600</a>
%H A341029 Agnes I. Benedek and Rafael Panzone, <a href="https://inmabb.criba.edu.ar/revuma/pdf/v39n1y2/p076-089.pdf">On Some Notable Plane Sets, II: Dragons</a>, Revista de la Unión Matemática Argentina, volume 39, numbers 1-2, 1994, pages 76-90.
%H A341029 Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index "HA".
%H A341029 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,3,2,-12,12,-12,8).
%F A341029 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.
%F A341029 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.
%F A341029 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) ).
%F A341029 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).
%e A341029   @   *---@          curve expansion level n=3,
%e A341029   |   |   |          convex hull vertices marked "@",
%e A341029   @---*   *---@      area = 4+1/2,
%e A341029               |      a(3) = 2*area = 9
%e A341029           @---@
%o A341029 (PARI) my(h=[22,29,22,31]); a(n) = if(n<2,n, (7<<n - h[n%4+1]<<(n\2-1))\3 + 1);
%Y A341029 Cf. A177057 (fractal hull area), A341030 (fractal hull perimeter).
%Y A341029 Cf. A362566 (bounding box area).
%K A341029 nonn,easy
%O A341029 0,3
%A A341029 _Kevin Ryde_, Feb 02 2021