A348920 a(n) is the "real" 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 A348921 gives "w" parts.

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

Offset

0,3

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 A334492 and of A348916.

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 A348920

Wikipedia, Eisenstein integer

Prog

(PARI) See Links section.

Crossrefs

Cf. A334492, A348916, A348921.

Sequence in context: A101664 A091952 A108803 * A348921 A050601 A101650

Adjacent sequences: A348917 A348918 A348919 * A348921 A348922 A348923

Keyword

sign,base

Author

Rémy Sigrist, Nov 04 2021

Status

approved