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A349008 Area of the convex hull around R5 dragon curve expansion level n.

%I A349008 #16 Dec 19 2024 11:46:19
%S A349008 0,2,16,106,578,2954,15064,75908,380334,1904330,9528618,47654840,
%T A349008 238295096,1191556256,5957936770,29789860126,148950201902,
%U A349008 744752899780,3723767329212,18618843605284,93094240114350,465471240742354,2327356261817746,11636781564585616
%N A349008 Area of the convex hull around R5 dragon curve expansion level n.
%C A349008 Expansion level n is the first 5^n segments of the curve and so the hull is around X,Y points located at X=A349195(t), Y=A349196(t) for t = 0..5^n.
%H A349008 Kevin Ryde, <a href="/A349008/b349008.txt">Table of n, a(n) for n = 0..400</a>
%H A349008 Kevin Ryde, <a href="http://user42.tuxfamily.org/r5dragon/index.html">Iterations of the R5 Dragon Curve</a>, see index "HA".
%H A349008 Kevin Ryde, <a href="/A349008/a349008.gp.txt">PARI/GP Code</a>
%F A349008 For n>=2, a(n) = 17*5^(n-2) - 1 + Sum_{j=1..n-2} ( (3*5^(n-2-j)-1)*HAgrow(2*b^j) + 2*5^(n-2-j)*HAgrow((4-i)*b^j) ),
%F A349008 where complex b=1+2*i and
%F A349008 HAgrow(z) = MinReIm(ShearRe(RotQ(z))),
%F A349008 MinReIm(z) = min(abs(Re z),abs(Im z)),
%F A349008 ShearRe(z) = z + Re(z),
%F A349008 RotQ(z) = z if sign(Re z) = sign(Im z) or RotQ(z) = z*i otherwise.
%e A349008 For n=2 the curve is:
%e A349008   @--@
%e A349008   |
%e A349008   *--*  *--*  *--@      Hull vertices "@".
%e A349008      |  |  |  |  |      Hull area a(2) = 16.
%e A349008   *--*--*--*--*--*
%e A349008   |  |  |  |  |
%e A349008   @--*  *--*  *--*
%e A349008                  |
%e A349008               @--@
%o A349008 (PARI) \\ See links.
%Y A349008 Cf. A349195, A349196 (coordinates).
%Y A349008 Cf. A349009 (fractal area).
%K A349008 nonn,easy
%O A349008 0,2
%A A349008 _Kevin Ryde_, Nov 06 2021