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

0, 2, 16, 106, 578, 2954, 15064, 75908, 380334, 1904330, 9528618, 47654840, 238295096, 1191556256, 5957936770, 29789860126, 148950201902, 744752899780, 3723767329212, 18618843605284, 93094240114350, 465471240742354, 2327356261817746, 11636781564585616

Offset: 0

Author

Kevin Ryde, Nov 06 2021

Keywords

• nonn
• easy

Comments

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.

Examples

			For n=2 the curve is:
  @--@
  |
  *--*  *--*  *--@      Hull vertices "@".
     |  |  |  |  |      Hull area a(2) = 16.
  *--*--*--*--*--*
  |  |  |  |  |
  @--*  *--*  *--*
                 |
              @--@
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..400
• Kevin Ryde, Iterations of the R5 Dragon Curve, see index "HA".
• Kevin Ryde, PARI/GP Code

Crossrefs

Cf. A349195, A349196 (coordinates).

Cf. A349009 (fractal area).

Programs

PARI

\\ See links.

Formula

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) ),

where complex b=1+2*i and

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

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

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

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