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

Author

Kevin Ryde, Nov 06 2021

Keywords

• cons
• nonn

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.

Examples

			0.97616400291270351340640715808421112...
		

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

Crossrefs

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

Programs

PARI

\\ See links.

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.