A293207 -id:A293207 - cp's OEIS frontend

A293539 Let P be the sequence of distinct lattice points defined by the following rules: P(1) = (0,0), P(2) = (1,0), and for any n > 2, P(n) is the closest lattice point to P(n-1) such that the angle of the vectors (P(n-2), P(n-1)) and (P(n-1), P(n)), say t, satisfies 0 < t <= Pi/2, and in case of a tie, minimize the angle t; a(n) = X-coordinate of P(n).

0, 1, 1, 0, -1, -1, 0, 2, 2, 1, 0, -1, -2, -2, -1, 1, 3, 3, 2, 2, 3, 3, 2, 1, -1, -2, -2, 0, 1, 4, 4, 3, 2, 1, 0, -3, -3, -2, -1, 2, 5, 5, 4, 4, 5, 5, 4, 3, 2, 1, 0, -3, -4, -4, -3, -3, -4, -4, -3, -2, 0, -2, -3, -5, -5, -4, 0, 1, 1, -5, -5, -4, -4, -5, -6, -6

Offset: 1

Rémy Sigrist, Oct 11 2017

See A293540 for the Y-coordinate of P(n).
The following diagram depicts the angle t cited in the name:
.      P(n)*    .
.          | t .
.          |  .
.          | .
.          |.
.    P(n-1)*
.         /
.        /
. P(n-2)*
The sequence P has similarities with Langton's ant:
- after an apparently chaotic initial phase, an escape consisting of a repetitive pattern emerges at n = 9118 (see illustrations in Links section),
- more formally: P(n+258) = P(n) + (14,-8) for any n >= 9118,
- See A274369 and A274370 for the coordinates of Langton's ant,
- See also A293207 for other sequences of points with emerging escapes.
See also A292469 for a sequence of points with similar construction features.

			See representation of first points in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..12000
Wikipedia, Langton's ant
Rémy Sigrist, Representation of P(n) for n=1..42, with lines joining consecutive points
Rémy Sigrist, Representation of P(n) for n=1..500, with lines joining consecutive points
Rémy Sigrist, Representation of the repetitive pattern emerging at n=9118
Rémy Sigrist, Colorized representation of the points P(n) for n=1..12000
Rémy Sigrist, Colorized representation of the points P'(n) of the variant where we maximize the angle t in case of a tie for n=1..1000000
Rémy Sigrist, PARI program for A293539

Cf. A274369, A274370, A292469, A293207, A293540.

PARI
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See Links section.
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a(n + 258) = a(n) + 14 for any n >= 9118.

A293208 Let b be the lexicographically earliest sequence of positive terms such that the function f defined by f(n) = Sum_{k=1..n} (i^k * b(k)) for any n >= 0 is injective (where i denotes the imaginary unit), and b(n) != b(n+1) and b() != b(n+2) for any n > 0; a(n) = real part of f(n).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Oct 02 2017

See A293207 for the corresponding sequence b, and additional comments.

			f(0) = 0, and a(0) = 0.
f(2) = f(1) + (i^1) * A293207(1) = 0 + (i) * 1 = i, and a(1) = 0.
f(3) = f(2) + (i^2) * A293207(2) = i + (-1) * 2 = -2 + i, and a(2) = -2.
f(4) = f(3) + (i^3) * A293207(3) = -2 + i + (-i) * 3 = -2 - 2*i, and a(3) = -2.
f(5) = f(4) + (i^4) * A293207(4) = -2 - 2*i + (1) * 1 = -1 - 2*i, and a(4) = -1.
f(6) = f(5) + (i^5) * A293207(5) = -1 - 2*i + (i) * 2 = -1, and a(5) = -1.
f(7) = f(6) + (i^6) * A293207(6) = -1 + (-1) * 3 = -4, and a(6) = -4.
f(8) = f(7) + (i^7) * A293207(7) = -4 + (-i) * 1 = -4 - i, and a(7) = -4.
f(9) = f(8) + (i^8) * A293207(8) = -4 - i + (1) * 2 = -2 - i, and a(8) = -2.
f(10) = f(9) + (i^9) * A293207(9) = -2 - i + (i) * 3 = -2 + 2*i, and a(9) = -2.
f(11) = f(10) + (i^10) * A293207(10) = -2 + 2*i + (-1) * 1 = -3 + 2*i, and a(10) = -3.

Rémy Sigrist, Table of n, a(n) for n = 0..59999
Rémy Sigrist, PARI program for A293208

Cf. A293207.

PARI
```
See Links section.
```

A293209 Let b be the lexicographically earliest sequence of positive terms such that the function f defined by f(n) = Sum_{k=1..n} (i^k * b(k)) for any n >= 0 is injective (where i denotes the imaginary unit), and b(n) != b(n+1) and b() != b(n+2) for any n > 0; a(n) = imaginary part of f(n).

Original entry on oeis.org

0, 1, 1, -2, -2, 0, 0, -1, -1, 2, 2, 0, 0, 1, 1, -3, -3, -1, -1, -2, -2, 3, 3, -4, -4, -3, -3, -5, -5, -1, -1, -2, -2, 1, 1, -1, -1, 2, 2, -2, -2, -1, -1, -3, -3, 0, 0, -1, -1, 1, 1, -5, -5, -4, -4, -6, -6, 0, 0, -7, -7, -4, -4, -5, -5, 4, 4, 2, 2, 3, 3, -4

Offset: 0

Rémy Sigrist, Oct 02 2017

sign

See A293207 for the corresponding sequence b, and additional comments.

			f(0) = 0, and a(0) = 0.
f(2) = f(1) + (i^1) * A293207(1) = 0 + (i) * 1 = i, and a(1) = 1.
f(3) = f(2) + (i^2) * A293207(2) = i + (-1) * 2 = -2 + i, and a(2) = 1.
f(4) = f(3) + (i^3) * A293207(3) = -2 + i + (-i) * 3 = -2 - 2*i, and a(3) = -2.
f(5) = f(4) + (i^4) * A293207(4) = -2 - 2*i + (1) * 1 = -1 - 2*i, and a(4) = -2.
f(6) = f(5) + (i^5) * A293207(5) = -1 - 2*i + (i) * 2 = -1, and a(5) = 0.
f(7) = f(6) + (i^6) * A293207(6) = -1 + (-1) * 3 = -4, and a(6) = 0.
f(8) = f(7) + (i^7) * A293207(7) = -4 + (-i) * 1 = -4 - i, and a(7) = -1.
f(9) = f(8) + (i^8) * A293207(8) = -4 - i + (1) * 2 = -2 - i, and a(8) = -1.
f(10) = f(9) + (i^9) * A293207(9) = -2 - i + (i) * 3 = -2 + 2*i, and a(9) = 2.
f(11) = f(10) + (i^10) * A293207(10) = -2 + 2*i + (-1) * 1 = -3 + 2*i, and a(10) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..59999
Rémy Sigrist, PARI program for A293209

Cf. A293207.

PARI
```
See Links section.
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A293208 Let b be the lexicographically earliest sequence of positive terms such that the function f defined by f(n) = Sum_{k=1..n} (i^k * b(k)) for any n >= 0 is injective (where i denotes the imaginary unit), and b(n) != b(n+1) and b() != b(n+2) for any n > 0; a(n) = real part of f(n).

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PARI

A293209 Let b be the lexicographically earliest sequence of positive terms such that the function f defined by f(n) = Sum_{k=1..n} (i^k * b(k)) for any n >= 0 is injective (where i denotes the imaginary unit), and b(n) != b(n+1) and b() != b(n+2) for any n > 0; a(n) = imaginary part of f(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI