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

0,2

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.

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

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.

Example

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

Prog

(PARI) \\ See links.

Crossrefs

Cf. A349195, A349196 (coordinates).

Cf. A349009 (fractal area).

Sequence in context: A236958 A009619 A012024 * A366635 A193217 A329550

Adjacent sequences: A349005 A349006 A349007 * A349009 A349010 A349011

Keyword

nonn,easy,changed

Author

Kevin Ryde, Nov 06 2021

Status

approved