A349010 Decimal expansion of the perimeter of the convex hull around the R5 dragon fractal.

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

Offset: 1

Author

Kevin Ryde, Nov 06 2021

Keywords

• cons
• nonn

Comments

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

With complex b = 1+2i, the hull sides are a countably infinite set: +-(4-i)/b^2, +-2/b^2, and 2*i^d/b^k for d=0..3 and k>=3. The sum of their magnitudes is the present constant.

Examples

			3.7436694412469800984922334098821413...
		

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• Kevin Ryde, Iterations of the R5 Dragon Curve, see index "HBf".

Crossrefs

Cf. A349009 (area).

Programs

Mathematica

RealDigits[(6 + 2*Sqrt[5] + 2*Sqrt[17])/5, 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

PARI

my(c=352+32*quadgen(5*17*4)); a_vector(len) = my(s=10^(len-2)); digits(sqrtint(floor(c*s^2)) + floor(12*s));

Formula

Equals (6 + 2*sqrt(5) + 2*sqrt(17)) / 5.

Equals (sqrt(8*sqrt(5*17) + 88) + 6) / 5.

Largest root of 625*x^4 - 3000*x^3 + 1000*x^2 + 6240*x - 2736 = 0 (all roots are real).