A322574 -id:A322574 - cp's OEIS frontend

A322510 a(1) = 0, and for any n > 0, a(2n) = a(n) + k(n) and a(2n+1) = a(n) - k(n) where k(n) is the least positive integer not leading to a duplicate term in sequence a.

0, 1, -1, 4, -2, 2, -4, 5, 3, 6, -10, 7, -3, 8, -16, 15, -5, 12, -6, 19, -7, -9, -11, 22, -8, 9, -15, 28, -12, -14, -18, 16, 14, 10, -20, 13, 11, 17, -29, 20, 18, 21, -35, 23, -41, 24, -46, 57, -13, 26, -42, 35, -17, 25, -55, 29, 27, 30, -54, 31, -59, 32, -68

Offset: 1

Rémy Sigrist, Dec 13 2018

The point is that the same k(n) must be used for both a(2*n) and a(2*n+1). - N. J. A. Sloane, Dec 17 2019

Apparently every signed integer appears in the sequence.

			The first terms, alongside k(n) and associate children, are:
  n   a(n)  k(n)  a(2*n)  a(2*n+1)
  --  ----  ----  ------  --------
   1     0     1       1        -1
   2     1     3       4        -2
   3    -1     3       2        -4
   4     4     1       5         3
   5    -2     8       6       -10
   6     2     5       7        -3
   7    -4    12       8       -16
   8     5    10      15        -5
   9     3     9      12        -6
  10     6    13      19        -7

Rémy Sigrist, Table of n, a(n) for n = 1..10000

For k(n) see A330337, A330338.

See A305410, A304971 and A322574-A322575 for similar sequences.

lista(nn) = my (a=[0], s=Set(0)); for (n=1, ceil(nn/2), for (k=1, oo, if (!setsearch(s, a[n]+k) && !setsearch(s, a[n]-k), a=concat(a, [a[n]+k, a[n]-k]); s=setunion(s, Set([a[n]+k, a[n]-k])); break))); a[1..nn]

a(n) = (a(2*n) + a(2*n+1))/2.

A322575 z(1) = 0, and for any n > 0, z(4n-2) = z(n) + k(n), z(4n-1) = z(n) + ik(n), z(4n) = z(n) - k(n) and z(4n+1) = z(n) - ik(n) where k(n) is the least positive integer not leading to a duplicate term in sequence z (and i denotes the imaginary unit); a(n) is the imaginary part of z(n).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Dec 17 2018

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A322575

See A322574 for the real part of z and additional comments.

PARI
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\\ See Links section.
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A326280 Let f(n) be a sequence of distinct Gaussian integers such that f(1) = 0 and for any n > 1, f(n) = f(floor(n/2)) + k(n)g((1+i)^(A000120(n)-1) (1-i)^A023416(n)) where k(n) > 0 is as small as possible and g(z) = z/gcd(Re(z), Im(z)); a(n) is the real part of f(n).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jun 22 2019

The idea underlying this sequence is to build an infinite binary tree of Gaussian integers:
- for any n > 0, f(n) has children f(2*n) and f(2*n+1),
- f(n), f(2*n) and f(2*n+1) form a right triangle,
- when u has child v and v has child w, then the angle between the vectors (u,v) and (v,w) is 45 degrees.
Among the first 2^20-1 terms, some values around the origin are missing: -2 - 3*i, -2, i, 2 - 2*i, 2, 4 + i, 5 - 2*i; will they ever appear?
Graphically, f has interesting features (see representations of f in Links section).
This sequence has similarities with A322574.

			See representation of the first layers of the binary tree in links section.

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, Representation of the first layers of the binary tree
Rémy Sigrist, Colored representation of f(n) for n = 1..2^20-1 (where the hue is function of n)
Rémy Sigrist, Colored representation of f(n) for n = 1..2^20-1 (where black pixels correspond to even n)
Rémy Sigrist, Density plot of the first 2^22-1 terms
Rémy Sigrist, PARI program for A326280

See A326281 for the imaginary part of f.

Cf. A000120, A023416, A322574.

PARI
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See Links section.
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cp's OEIS Frontend

A322510 a(1) = 0, and for any n > 0, a(2n) = a(n) + k(n) and a(2n+1) = a(n) - k(n) where k(n) is the least positive integer not leading to a duplicate term in sequence a.

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A322575 z(1) = 0, and for any n > 0, z(4n-2) = z(n) + k(n), z(4n-1) = z(n) + ik(n), z(4n) = z(n) - k(n) and z(4n+1) = z(n) - ik(n) where k(n) is the least positive integer not leading to a duplicate term in sequence z (and i denotes the imaginary unit); a(n) is the imaginary part of z(n).

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PARI

A326280 Let f(n) be a sequence of distinct Gaussian integers such that f(1) = 0 and for any n > 1, f(n) = f(floor(n/2)) + k(n)g((1+i)^(A000120(n)-1) (1-i)^A023416(n)) where k(n) > 0 is as small as possible and g(z) = z/gcd(Re(z), Im(z)); a(n) is the real part of f(n).

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Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

cp's OEIS Frontend

A322510 a(1) = 0, and for any n > 0, a(2*n) = a(n) + k(n) and a(2*n+1) = a(n) - k(n) where k(n) is the least positive integer not leading to a duplicate term in sequence a.

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PARI

Formula

A322575 z(1) = 0, and for any n > 0, z(4*n-2) = z(n) + k(n), z(4*n-1) = z(n) + i*k(n), z(4*n) = z(n) - k(n) and z(4*n+1) = z(n) - i*k(n) where k(n) is the least positive integer not leading to a duplicate term in sequence z (and i denotes the imaginary unit); a(n) is the imaginary part of z(n).

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A326280 Let f(n) be a sequence of distinct Gaussian integers such that f(1) = 0 and for any n > 1, f(n) = f(floor(n/2)) + k(n)*g((1+i)^(A000120(n)-1) * (1-i)^A023416(n)) where k(n) > 0 is as small as possible and g(z) = z/gcd(Re(z), Im(z)); a(n) is the real part of f(n).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A322510 a(1) = 0, and for any n > 0, a(2n) = a(n) + k(n) and a(2n+1) = a(n) - k(n) where k(n) is the least positive integer not leading to a duplicate term in sequence a.

A322575 z(1) = 0, and for any n > 0, z(4n-2) = z(n) + k(n), z(4n-1) = z(n) + ik(n), z(4n) = z(n) - k(n) and z(4n+1) = z(n) - ik(n) where k(n) is the least positive integer not leading to a duplicate term in sequence z (and i denotes the imaginary unit); a(n) is the imaginary part of z(n).

A326280 Let f(n) be a sequence of distinct Gaussian integers such that f(1) = 0 and for any n > 1, f(n) = f(floor(n/2)) + k(n)g((1+i)^(A000120(n)-1) (1-i)^A023416(n)) where k(n) > 0 is as small as possible and g(z) = z/gcd(Re(z), Im(z)); a(n) is the real part of f(n).