A334493 a(n) is the "w" part of f(n) = Sum_{k>=0, d_k>0} (1+w)^(d_k-1) * (3+w)^k where Sum_{k>=0} d_k * 7^k is the base 7 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A334492 gives "real" parts.

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

Offset: 0

Author

Rémy Sigrist, May 03 2020

Keywords

• sign
• base
• look

Comments

For any Eisenstein integer z = u + v*w (where u and v are integers), we call u the "real" part of z and v the "w" part of z.

This sequence has connections with A316658; here we work with Eisenstein integers, there with Gaussian integers.

It appears that f defines a bijection from the nonnegative integers to the Eisenstein integers.

Examples

			The following diagram depicts f(n) for n = 0..13:
            "w" axis
                \
           .     .     .     .     .     .     .     .
                  \              10     9
                   \
        .     .     .     .     .     .     .     .
                   3 \   2    11     7     8
                      \
           ._____._____._____._____._____._____._____. "real" axis
                4     0 \   1    12    13
                         \
        .     .     .     .     .     .     .     .
                   5     6 \
- f(9) = 4 + 2*w, hence a(9) = 2.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..16806
• Rémy Sigrist, PARI program for A334493
• Wikipedia, Eisenstein integer

Crossrefs

Cf. A307012 (equivalent coordinate for a counterclockwise spiral), A316658, A334492.

Programs

PARI

See Links section.