A349008 - OEIS

%I #15 Nov 17 2021 08:07:07

%S 0,2,16,106,578,2954,15064,75908,380334,1904330,9528618,47654840,

%T 238295096,1191556256,5957936770,29789860126,148950201902,

%U 744752899780,3723767329212,18618843605284,93094240114350,465471240742354,2327356261817746,11636781564585616

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

%C 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 Kevin Ryde, <a href="/A349008/b349008.txt">Table of n, a(n) for n = 0..400</a>

%H Kevin Ryde, <a href="http://user42.tuxfamily.org/r5dragon/index.html">Iterations of the R5 Dragon Curve</a>, see index "HA".

%H Kevin Ryde, <a href="/A349008/a349008.gp.txt">PARI/GP Code</a>

%F 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 where complex b=1+2*i and

%F HAgrow(z) = MinReIm(ShearRe(RotQ(z))),

%F MinReIm(z) = min(abs(Re z),abs(Im z)),

%F ShearRe(z) = z + Re(z),

%F RotQ(z) = z if sign(Re z) = sign(Im z) or RotQ(z) = z*i otherwise.

%e For n=2 the curve is:

%e   @--@

%e   |

%e   *--*  *--*  *--@      Hull vertices "@".

%e      |  |  |  |  |      Hull area a(2) = 16.

%e   *--*--*--*--*--*

%e   |  |  |  |  |

%e   @--*  *--*  *--*

%e                  |

%e               @--@

%o (PARI) See links.

%Y Cf. A349195, A349196 (coordinates).

%Y Cf. A349009 (fractal area).

%K nonn,easy

%O 0,2

%A _Kevin Ryde_, Nov 06 2021