A329981 -id:A329981 - cp's OEIS frontend

A329985 a(1) = 1 and for n > 0, a(n+1) = a(k) - a(n) where k is the number of terms equal to a(n) among the first n terms.

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

Offset: 1

Rémy Sigrist, Nov 26 2019

In other words, for n > 0, a(n+1) = a(o(n)) - a(n) where o is the ordinal transform of the sequence.

The sequence has interesting graphical features (see plot in Links section).

			The first terms, alongside their ordinal transform, are:
  n   a(n)  o(n)
  --  ----  ----
   1     1     1
   2     0     1
   3     1     2
   4    -1     1
   5     2     1
   6    -1     2
   7     1     3
   8     0     2
   9     0     3
  10     1     4

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, Density plot of the first 2^22 terms
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

o(n) is A330334.

See A329981 for similar sequences.

A={1};For[n=2,n<=76,n++,A=Append[A,Part[A,Count[Table[Part[A,i],{i,1,n-1}],Part[A,n-1]]]-Part[A,n-1]]];A (* Joshua Oliver, Nov 26 2019 *)
Nest[Append[#, #[[Count[#, #[[-1]] ] ]] - #[[-1]]] &, {1}, 75] (* Michael De Vlieger, Dec 01 2019 *)

for (n=1, #(a=vector(76)), print1 (a[n]=if (n==1, 1, a[sum(k=1, n-1, a[k]==a[n-1])]-a[n-1])", "))

A329982 a(1) = 0, and for n > 0, a(n+1) = k^2 - a(n) where k is the number of terms equal to a(n) among the first n terms.

Original entry on oeis.org

0, 1, 0, 4, -3, 4, 0, 9, -8, 9, -5, 6, -5, 9, 0, 16, -15, 16, -12, 13, -12, 16, -7, 8, -7, 11, -10, 11, -7, 16, 0, 25, -24, 25, -21, 22, -21, 25, -16, 17, -16, 20, -19, 20, -16, 25, -9, 10, -9, 13, -9, 18, -17, 18, -14, 15, -14, 18, -9, 25, 0, 36, -35, 36, -32

Offset: 1

Rémy Sigrist, Nov 26 2019

In other words, for n > 0, a(n+1) = o(n)^2 - a(n) where o is the ordinal transform of the sequence.

			The first terms, alongside their ordinal transform, are:
  n   a(n)  o(n)
  --  ----  ----
   1     0     1
   2     1     1
   3     0     2
   4     4     1
   5    -3     1
   6     4     2
   7     0     3
   8     9     1
   9    -8     1
  10     9     2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 2^20 terms

See A329981 for similar sequences.

for (n=1, #(a=vector(65)), print1 (a[n]=if (n>1, sum(k=1, n-1, a[k]==a[n-1])^2-a[n-1])", "))

A330004 a(1) = 0, and for n > 0, a(n+1) = u - v where u (resp. v) is the number of terms equal to a(n) (resp. a(n)+1) among the first n terms.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Nov 26 2019

The sequence has chaotic features (see plot in Links section).

			The first terms, alongside u and v, are:
  n   a(n)  u  v
  --  ----  -  -
   1     0  1  0
   2     1  1  0
   3     1  2  0
   4     2  1  0
   5     1  3  1
   6     2  2  0
   7     2  3  0
   8     3  1  0
   9     1  4  3
  10     1  5  3

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Density plot of the first 25000000 terms

See A329981 for similar sequences.

for (n=1, #(a=vector(85)), print1 (a[n]=if (n==1, 0, sum(k=1, n-1, (a[k]==a[n-1])-(a[k]==a[n-1]+1)))", "))

A329984 a(1) = 0 and for n > 0, a(n+1) is the odd part of k where k is the number of terms equal to a(n) among the first n terms.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 9, 1, 5, 1, 11, 1, 3, 3, 1, 13, 1, 7, 1, 15, 1, 1, 17, 1, 9, 1, 19, 1, 5, 3, 5, 1, 21, 1, 11, 1, 23, 1, 3, 3, 7, 3, 1, 25, 1, 13, 1, 27, 1, 7, 1, 29, 1, 15, 1, 31, 1, 1, 33, 1, 17, 1, 35, 1, 9, 3, 9, 1, 37, 1, 19, 1

Offset: 1

Rémy Sigrist, Nov 26 2019

In other words, for n > 0, a(n+1) = A000265(o(n)) where o is the ordinal transform of the sequence.

			The first terms, alongside their ordinal transform, are:
  n   a(n)  o(n)
  --  ----  ----
   1     0     1
   2     1     1
   3     1     2
   4     1     3
   5     3     1
   6     1     4
   7     1     5
   8     5     1
   9     1     6
  10     3     2

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the first 2^16 terms

See A329981 for similar sequences.

Cf. A000265.

o=vector(38); v=0; for (n=1, 80, print1 (v", "); o[1+v]++; v=o[1+v]/2^valuation(o[1+v],2))

cp's OEIS Frontend

A329985 a(1) = 1 and for n > 0, a(n+1) = a(k) - a(n) where k is the number of terms equal to a(n) among the first n terms.

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A329982 a(1) = 0, and for n > 0, a(n+1) = k^2 - a(n) where k is the number of terms equal to a(n) among the first n terms.

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A330004 a(1) = 0, and for n > 0, a(n+1) = u - v where u (resp. v) is the number of terms equal to a(n) (resp. a(n)+1) among the first n terms.

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Examples

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Crossrefs

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PARI

A329984 a(1) = 0 and for n > 0, a(n+1) is the odd part of k where k is the number of terms equal to a(n) among the first n terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI