A348917 -id:A348917

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A348916

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) = (2 + w)^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 A348917 gives "w" parts.

+20
3

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

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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 combines features of A334492 and of A348652.

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

. .

\ 4

. . . .

6 5 \ 3 2

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

7 0 \ 1

. . . .

8 9 11 \ 12

. .

10 \

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 A348916

Wikipedia, Eisenstein integer

PROG

(PARI) See Links section.

CROSSREFS

Cf. A334492, A348652, A348917.

KEYWORD

sign,base

AUTHOR

Rémy Sigrist, Nov 03 2021

STATUS

approved

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.

+10
2

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