A349009 Decimal expansion of the area of the convex hull around the R5 dragon fractal.

9, 7, 6, 1, 6, 4, 0, 0, 2, 9, 1, 2, 7, 0, 3, 5, 1, 3, 4, 0, 6, 4, 0, 7, 1, 5, 8, 0, 8, 4, 2, 1, 1, 1, 2, 9, 7, 2, 6, 3, 1, 2, 1, 9, 9, 3, 1, 7, 3, 2, 6, 9, 0, 5, 2, 4, 3, 4, 9, 4, 8, 8, 0, 3, 0, 0, 8, 2, 8, 7, 3, 8, 6, 7, 9, 6, 5, 1, 1, 6, 0, 1, 1, 0, 7, 5, 0, 4, 2, 4, 7, 8, 8, 5, 1, 6, 1, 5, 8, 6, 3, 8, 6, 6, 9

Offset

0,1

Comments

The fractal is taken scaled to unit length from curve start to end.

In the sum formula below, all HAtermf(j) > 0 and a simple upper bound is Sum_{j>=k} HAtermf(j) < 1/sqrt(5)^k.

Links

Kevin Ryde, Table of n, a(n) for n = 0..10000

Kevin Ryde, Iterations of the R5 Dragon Curve, see index "HAf".

Kevin Ryde, PARI/GP Code

Formula

Equals 17/25 + Sum_{j>=1} HAtermf(j), where complex b=1+2*i and:

HAtermf(j) = (1/25)*(6*HAgrowf(1/b^j) + 2*HAgrowf((4+i)/b^j)),

HAgrowf(z) = MinReIm(ShearIm(RotQ(z))),

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

ShearIm(z) = z + i*Im(z),

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

Equals lim_{n->oo} A349008(n)/5^n.

Example

0.97616400291270351340640715808421112...

Prog

(PARI) See links.

Crossrefs

Cf. A349008 (finite areas), A349010 (fractal perimeter).

Sequence in context: A194554 A065467 A021839 * A094131 A021510 A199272

Adjacent sequences: A349006 A349007 A349008 * A349010 A349011 A349012

Keyword

cons,nonn

Author

Kevin Ryde, Nov 06 2021

Status

approved