A332249 a(n) is the X-coordinate of the n-th point of the quadratic Koch curve. Sequence A332250 gives Y-coordinates.

0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 6, 6, 7, 7, 8, 8, 9, 9, 8, 8, 9, 9, 8, 8, 7, 7, 6, 6, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 15, 15, 16, 16, 17, 17, 18, 18, 19

Offset: 0

Author

Rémy Sigrist, Feb 08 2020

Keywords

• nonn
• base

Comments

This sequence is the real part of {f(n)} defined as:

- f(0) = 0,

- f(n+1) = f(n) + i^t(n)

where t(n) is the number of 1's minus the number of 3's

in the base 5 representation of n

and i denotes the imaginary unit.

We can also build the curve by successively applying the following substitution to an initial vector (1, 0):

.--->.

^ |

| v

.--->. .--->.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..15625
• Robert Ferréol (MathCurve), Courbe de Koch quadratique [in French]
• Rémy Sigrist, Representation of the first 1+2^10 points of the quadratic Koch curve
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

See A332246 for a similar sequence.

Cf. A229217, A332250 (Y-coordinates).

Programs

PARI

{ k = [0, 1, 0, -1, 0]; z=0; for (n=0, 77, print1 (real(z) ", "); z += I^vecsum(apply(d -> k[1+d], digits(n, #k)))) }

Formula

a(5^k-m) + a(m) = 3^k for any k >= 0 and m = 0..5^k.