A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k*13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)*(3+2*i)^k where g(0) = 0, and g(1+u+3*v) = (1+u*i)*i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

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

Offset: 0

Author

Rémy Sigrist, Oct 27 2021

Keywords

• sign
• base

Comments

The function f defines a bijection from the nonnegative integers to the Gaussian integers.

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

|

| +

| 3

|

+ + + +

6 5 |4 2

|

--------+----+----+-------

7 |0 1

|

+ + + +

8 |10 11 12

|

+ |

9 |

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..2197
• Stephen K. Lucas, Base 2 + i with digit set {0, +/-1, +/-i}, ResearchGate (October 2021).
• Rémy Sigrist, Colored representation of f for n = 0..13^5-1 in the complex plane (the hue is function of n)

Crossrefs

See A316658 for a similar sequence.

Cf. A188982, A348652.

Programs

PARI

g(d) = { if (d==0, 0, (1+I*((d-1)%3))*I^((d-1)\3)) }
a(n) = imag(subst(Pol([g(d)|d<-digits(n, 13)]), 'x, 3+2*I))

Formula

abs(a(13^k)) = A188982(k).