A318706 -id:A318706 - cp's OEIS frontend

A318705 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2j) = i^j and s(2+2j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the real part of g(n).

0, 1, 1, 0, -1, -1, -1, 0, 1, 3, 4, 4, 3, 2, 2, 2, 3, 4, 3, 4, 4, 3, 2, 2, 2, 3, 4, 0, 1, 1, 0, -1, -1, -1, 0, 1, -3, -2, -2, -3, -4, -4, -4, -3, -2, -3, -2, -2, -3, -4, -4, -4, -3, -2, -3, -2, -2, -3, -4, -4, -4, -3, -2, 0, 1, 1, 0, -1, -1, -1, 0, 1, 3, 4, 4

Offset: 0

Rémy Sigrist, Sep 01 2018

See A318706 for the imaginary part of g.
See A318707 for the square of the modulus of g.
The following diagrams shows s(k) for k = 0..8 in the complex plane:
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     s(4)  s(3)  s(2)
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  ---s(5)--s(0)--s(1)---
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     s(6)  s(7)  s(8)
            |
            |
The function g defines a bijection from the nonnegative integers to the Gaussian integers.
This sequence has similarities with A316657.

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, Colored scatterplot of (a(n), A318706(n)) for n = 0..9^6-1 (where the hue is function of n)

Cf. A316657, A318706, A318707.

a(n) = my (d=Vecrev(digits(n, 9))); real(sum(k=1, #d, if (d[k], 3^(k-1)*I^floor((d[k]-1)/2)*(1+I)^((d[k]-1)%2), 0)))

a(9 * k) = 3 * a(k) for any k >= 0.

A318707 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2j) = i^j and s(2+2j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the square of the modulus of g(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 1, 2, 9, 16, 17, 10, 5, 4, 5, 10, 17, 18, 25, 32, 25, 20, 13, 8, 13, 20, 9, 10, 17, 16, 17, 10, 5, 4, 5, 18, 13, 20, 25, 32, 25, 20, 13, 8, 9, 4, 5, 10, 17, 16, 17, 10, 5, 18, 13, 8, 13, 20, 25, 32, 25, 20, 9, 10, 5, 4, 5, 10, 17, 16, 17

Offset: 0

Rémy Sigrist, Sep 01 2018

See A318705 for the real part of g and additional comments.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A318705, A318706.

a(n) = my (d=Vecrev(digits(n, 9))); norm(sum(k=1, #d, if (d[k], 3^(k-1)*I^floor((d[k]-1)/2)*(1+I)^((d[k]-1)%2), 0)))

a(n) = A318705(n)^2 + A318706(n)^2.

a(9 * k) = 9 * a(k) for any k >= 0.

cp's OEIS Frontend

A318705 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2j) = i^j and s(2+2j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the real part of g(n).

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A318707 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2j) = i^j and s(2+2j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the square of the modulus of g(n).

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cp's OEIS Frontend

A318705 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2*j) = i^j and s(2+2*j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the real part of g(n).

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A318707 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2*j) = i^j and s(2+2*j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the square of the modulus of g(n).

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PARI

Formula

A318705 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2j) = i^j and s(2+2j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the real part of g(n).

A318707 For any n >= 0 with base-9 representation Sum_{k=0..w} d_k * 9^k, let g(n) = Sum_{k=0..w} s(d_k) * 3^k (where s(0) = 0, s(1+2j) = i^j and s(2+2j) = i^j * (1+i) for any j > 0, and i denotes the imaginary unit); a(n) is the square of the modulus of g(n).