A330780 -id:A330780 - cp's OEIS frontend

A331004 a(n) is the real part of f(n) (as defined in A330780).

0, 2, 4, 2, -2, -6, -9, -6, 0, 6, 9, 6, 0, -8, -11, -8, 0, 8, 12, 8, 0, -8, -12, -7, 1, 9, 13, 9, 1, -7, -12, -8, 0, 10, 15, 10, 0, -10, -15, -10, 2, 12, 17, 12, 2, -10, -15, -10, 0, 10, 15, 10, -2, -12, -17, -12, -2, 8, 14, 9, -3, -15, -21, -15, -3, 9, 15, 8

Offset: 0

Rémy Sigrist, Jan 05 2020

sign

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A331004

Cf. A330780.

PARI
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See Links section.
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A331005 a(n) is the imaginary part of f(n) (as defined in A330780).

Original entry on oeis.org

0, 1, 5, 9, 11, 9, 3, -3, -6, -3, 3, 9, 12, 8, 2, -4, -8, -4, 4, 12, 16, 12, 4, -6, -10, -6, 2, 10, 14, 10, 0, -8, -12, -7, 3, 13, 18, 13, 3, -7, -13, -8, 2, 12, 17, 11, 1, -9, -14, -9, 1, 11, 17, 12, 2, -8, -13, -8, 4, 14, 20, 14, 2, -10, -16, -10, 2, 16, 22

Offset: 0

Rémy Sigrist, Jan 05 2020

sign

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A331005

Cf. A330780.

PARI
```
See Links section.
```

A330779 Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * i^k for n >= 0 is injective (where i denotes the imaginary unit).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Dec 31 2019

nonn

The variant of this sequence where each value can only appear up to once, twice or three times corresponds to A000027, A008619 and A008620 respectively.

Graphically, the representation of f resembles a windmill; the variant of f where we allow the value v to appear 3*v times resembles a butterfly (see illustrations in Links section).

			The first terms, alongside the corresponding values of f(n), are:
  n   a(n)  f(n)
  --  ----  ------
   0  N/A        0
   1     1       i
   2     2    -2+i
   3     2    -2-i
   4     3     1-i
   5     3   1+2*i
   6     3  -2+2*i
   7     4  -2-2*i
   8     4   2-2*i
   9     4   2+2*i
  10     5  -3+2*i
  11     4  -3-2*i
  12     6   3-2*i
See also illustration in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of first steps
Rémy Sigrist, Colored representation of f(n) for n = 0..1000000 in the complex plane (where the color is function of n)
Rémy Sigrist, Colored representation of the variant where the value v can appear up to 3*v times
Rémy Sigrist, Colored representation of the variant where the value v can appear up to A000265(v) times
Rémy Sigrist, Colored representation of the variant where the value v can appear up to prime(v) times
Rémy Sigrist, PARI program for A330779

See A331002 and A331003 for the real and imaginary parts of f, respectively.
See A330780 for another variant.
Cf. A000265.

PARI
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See Links section.
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cp's OEIS Frontend

A331004 a(n) is the real part of f(n) (as defined in A330780).

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PARI

A331005 a(n) is the imaginary part of f(n) (as defined in A330780).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A330779 Lexicographically earliest sequence of positive integers such that for any v > 0, the value v appears up to v times, and the associate function f defined by f(n) = Sum_{k = 1..n} a(k) * i^k for n >= 0 is injective (where i denotes the imaginary unit).

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PARI