A332412 -id:A332412 - cp's OEIS frontend

A332413 a(n) is the imaginary part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332412 gives real parts.

0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 3, 3, 4, 3, 2, 0, 0, 1, 0, -1, -3, -3, -2, -3, -4, 0, 0, 1, 0, -1, 0, 0, 1, 0, -1, 3, 3, 4, 3, 2, 0, 0, 1, 0, -1, -3, -3, -2, -3, -4, 9, 9, 10, 9, 8, 9, 9, 10, 9, 8, 12, 12, 13, 12, 11, 9, 9, 10, 9, 8, 6, 6, 7, 6, 5, 0, 0, 1, 0

Offset: 0

Rémy Sigrist, Feb 12 2020

			For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -9.

Rémy Sigrist, Table of n, a(n) for n = 0..15624

Cf. A031219, A289814, A332412 (real parts and additional comments).

a(n) = { my (d=Vecrev(digits(n,5))); imag(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

a(n) = 0 iff the n-th row of A031219 has neither 2's nor 4's.
a(5*n)   = 3*a(n).
a(5*n+1) = 3*a(n).
a(5*n+2) = 3*a(n) + 1.
a(5*n+3) = 3*a(n).
a(5*n+4) = 3*a(n) - 1.

A332497 a(n) = x(w+1) where x(0) = 0 and x(k+1) = 2^(k+1)-1-x(k) (resp. x(k)) when d_k = 1 (resp. d_k <> 1) and Sum_{k=0..w} d_k*3^k is the ternary representation of n. Sequence A332498 gives corresponding y's.

Original entry on oeis.org

0, 1, 0, 3, 2, 3, 0, 1, 0, 7, 6, 7, 4, 5, 4, 7, 6, 7, 0, 1, 0, 3, 2, 3, 0, 1, 0, 15, 14, 15, 12, 13, 12, 15, 14, 15, 8, 9, 8, 11, 10, 11, 8, 9, 8, 15, 14, 15, 12, 13, 12, 15, 14, 15, 0, 1, 0, 3, 2, 3, 0, 1, 0, 7, 6, 7, 4, 5, 4, 7, 6, 7, 0, 1, 0, 3, 2, 3, 0, 1

Offset: 0

Rémy Sigrist, Feb 14 2020

The representation of {(a(n), A332498(n))} is related to the T-square fractal (see illustration in Links section).
We can iteratively build the set {(a(n), A332498(n))} as follows:
- start with X_0 = {(0, 0)},
- for k = 0, 1, ..., X_{k+1} is obtained by adjoining to X_k:
     - an horizontally mirrored copy of X_k to the right,
     - and a vertically mirrored copy of X_k on the top,
- this corresponds to the following substitution:
                     .---.
         .---.       | V |
         | X |  -->  .---.---.
         .---.       | X | H |
                     .---.---.

			For n = 42:
- the ternary representation of 42 is "1120",
- x(0) = 0,
- x(1) = x(0) = 0 (as d_0 = 0 <> 1),
- x(2) = x(1) = 0 (as d_1 = 2 <> 1),
- x(3) = 2^3-1 - x(2) = 7 (as d_2 = 1),
- x(4) = 2^4-1 - x(3) = 8 (as d_3 = 1),
- hence a(42) = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, Representation of (a(n), A332498(n)) for n = 0..3^10-1
Rémy Sigrist, Interactive scatterplot of a 3D analog [Provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes]
Wikipedia, T-square (fractal)

See A332412 for a similar sequence.

Cf. A005823, A332498 (corresponding y's).

a(n) = { my (x=0, k=1); while (n, if (n%3==1, x=2^k-1-x); n\=3; k++); x }

a(n) = 0 iff n belongs to A005823.

cp's OEIS Frontend

A332413 a(n) is the imaginary part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332412 gives real parts.

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