A348921 a(n) is the "w" part of f(n) = Sum_{k >= 0} g(d_k) * (4 + w)^k where g(0) = 0 and g(1 + u + 2*v) = (1 + u) * (1 + w)^v for any u = 0..1 and v = 0..5, Sum_{k >= 0} d_k * 13^k is the base-13 representation of n and w = -1/2 + sqrt(-3)/2 is a primitive cube root of unity; sequence A348920 gives "real" parts.

0, 0, 0, 1, 2, 1, 2, 0, 0, -1, -2, -1, -2, 1, 1, 1, 2, 3, 2, 3, 1, 1, 0, -1, 0, -1, 2, 2, 2, 3, 4, 3, 4, 2, 2, 1, 0, 1, 0, 4, 4, 4, 5, 6, 5, 6, 4, 4, 3, 2, 3, 2, 8, 8, 8, 9, 10, 9, 10, 8, 8, 7, 6, 7, 6, 3, 3, 3, 4, 5, 4, 5, 3, 3, 2, 1, 2, 1, 6, 6, 6, 7, 8, 7

Offset: 0

Author

Rémy Sigrist, Nov 04 2021

Keywords

• sign
• base

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 is a variant of A334493 and of A348917.

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

The following diagram depicts g(d) for d = 0..12:

"w" axis

\

. . .

6 \ 4

\

. .

5 \ 3

\

.___._____.___._____._ "real" axis

8 7 0 \ 1 2

\

. .

9 11 \

\

. . .

10 12 \

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..2196
• Joerg Arndt, Representation of a similar construction
• Rémy Sigrist, Colored representation of f for n = 0..13^5-1 in the complex plane (the hue is function of n)
• Rémy Sigrist, PARI program for A348921
• Wikipedia, Eisenstein integer

Crossrefs

Cf. A334493, A348917, A348920.

Programs

PARI

See Links section.