A348652 -id:A348652 - cp's OEIS frontend

A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)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

Rémy Sigrist, Oct 27 2021

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   |

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)

See A316658 for a similar sequence.

Cf. A188982, A348652.

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))

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

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 + 2v) = (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.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Nov 03 2021

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       \

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

Cf. A334492, A348652, A348917.

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cp's OEIS Frontend

A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.

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cp's OEIS Frontend

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.

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A348653 For any nonnegative number n with base-13 expansion Sum_{k >= 0} d_k13^k, a(n) is the imaginary part of Sum_{k >= 0} g(d_k)(3+2i)^k where g(0) = 0, and g(1+u+3v) = (1+ui)i^v for any u = 0..2 and v = 0..3 (where i denotes the imaginary unit); see A348652 for the real part.