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).

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

0,10

Comments

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)

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

This sequence has similarities with A316657.

Links

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)

Formula

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

Prog

(PARI) 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)))

Crossrefs

Cf. A316657, A318706, A318707.

Sequence in context: A248158 A360268 A090283 * A334492 A306571 A019917

Adjacent sequences: A318702 A318703 A318704 * A318706 A318707 A318708

Keyword

sign,base

Author

Rémy Sigrist, Sep 01 2018

Status

approved