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

0,5

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

Prog

(PARI) See Links section.

Crossrefs

Cf. A334493, A348917, A348920.

Sequence in context: A091952 A108803 A348920 * A050601 A101650 A287411

Adjacent sequences: A348918 A348919 A348920 * A348922 A348923 A348924

Keyword

sign,base

Author

Rémy Sigrist, Nov 04 2021

Status

approved