cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A330263 Distinct values of A308551 in order of their appearance as n grows.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 6, 15, 23, 7, 19, 30, 47, 57, 20, 16, 8, 22, 35, 58, 73, 82, 100, 112, 123, 39, 29, 9, 25, 40, 69, 85, 98, 121, 138, 155, 179, 232, 247, 269, 282, 26, 44, 80, 17, 10, 28, 46, 81, 96, 111, 143, 163, 181, 207, 259, 279, 298, 332, 351, 371, 392
Offset: 1

Views

Author

Rémy Sigrist, Dec 07 2019

Keywords

Comments

This sequence is a permutation of the natural numbers.

Examples

			The first terms of A308551 and of this sequence are:
- A308551 = 1, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 5, 1, 12, ...
- a       = 1, 2,    3,          4,                      5,    12, ...

Links

Crossrefs

Cf. A308551.

Programs

  • PARI
    See Links section.

A330261 Start with an empty stack S; for n = 1, 2, 3, ..., interpret the binary representation of n from left to right as follows: in case of bit 1, push the number 1 on top of S, in case of bit 0, replace the two numbers on top of S, say u on top of v, with u-v; a(n) gives the number on top of S after processing n.

Original entry on oeis.org

1, 0, 1, -1, 1, 0, 1, -2, 1, 1, 1, -1, 1, 0, 1, -3, 1, 4, 1, 0, 1, 0, 1, -2, 1, 1, 1, -1, 1, 0, 1, -4, 1, 5, 1, 7, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -3, 1, 2, 1, 0, 1, 0, 1, -2, 1, 1, 1, -1, 1, 0, 1, -5, 1, 5, 1, -4, 1, 0, 1, 3, 1, -3, 1, 1, 1, 0, 1, -2, 1, -2
Offset: 1

Views

Author

Rémy Sigrist, Dec 07 2019

Keywords

Comments

This sequence is a variant of A308551.
After processing n, S has A268289(n) elements.
Every integer appears infinitely many times in the sequence:
- the effect of the binary string b(0) = "110" is to leave 0 on top of S,
- the effect of the binary string b(1) = "1" is to leave 1 on top of S,
- the effect of the binary string b(-1) = "11100" is to leave -1 on top of S,
- let "|" denote the binary concatenation,
- for any k > 0:
- the effect of b(k+1) = b(-1)|b(k)|"0" is to leave k+1 on top of S,
- the effect of b(-k-1) = b(1)|b(-k)|"0" is to leave -k-1 on top of S,
- for any k, for any n > 0, if the binary representation of n ends with b(k), then a(n) = k, QED,
- see A330264 for the values in order of appearance.

Examples

			The first terms, alongside the binary representation of n and the evolution of stack S, are:
  n   a(n)  bin(n)  S
  --  ----  ------  ------------------------------------------------------------
   1     1       1  () -> (1)
   2     0      10  (1) -> (1,1) -> (0)
   3     1      11  (0) -> (0,1) -> (0,1,1)
   4    -1     100  (0,1,1) -> (0,1,1,1) -> (0,1,0) -> (0,-1)
   5     1     101  (0,-1) -> (0,-1,1) -> (0,2) -> (0,2,1)
   6     0     110  (0,2,1) -> (0,2,1,1) -> (0,2,1,1,1) -> (0,2,1,0)
   7     1     111  (0,2,1,0) -> (0,2,1,0,1) -> (0,2,1,0,1,1) -> (0,2,1,0,1,1,1)

Links

Crossrefs

Cf. A268289, A308551, A330261, A330264.

Programs

  • PARI
    See Links section.

Formula

a(2*k-1) = 1 for any k > 0.

A330262 Start with an empty stack S; for n = 1, 2, 3, ..., interpret the binary representation of n from left to right as follows: in case of bit 1, push the number 1 on top of S, in case of bit 0, replace the two numbers on top of S, say u on top of v, with v-u; a(n) gives the number on top of S after processing n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 0, 1, -1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, -1, 1, 1, 1, 0, 1, 0, 1, -1, 1, -1, 1, 0, 1, -1, 1, -2, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, -1, 1, 1, 1, 0, 1, 1, 1, -1, 1, 0, 1, 0, 1, -3, 1, -3, 1, 1, 1, 0, 1, 2, 1, -2, 1
Offset: 1

Views

Author

Rémy Sigrist, Dec 07 2019

Keywords

Comments

This sequence is a variant of A330261.
After processing n, S has A268289(n) elements.
Every integer appears infinitely many times in the sequence:
- the proof is similar to that found in A330261,
- see A330265 for the values in order of appearance.

Examples

			The first terms, alongside the binary representation of n and the evolution of stack S, are:
  n  a(n)  bin(n)  S
  -  ----  ------  ------------------------------------------------------------
  1     1       1  () -> (1)
  2     0      10  (1) -> (1,1) -> (0)
  3     1      11  (0) -> (0,1) -> (0,1,1)
  4     1     100  (0,1,1) -> (0,1,1,1) -> (0,1,0) -> (0,1)
  5     1     101  (0,1) -> (0,1,1) -> (0,0) -> (0,0,1)
  6     0     110  (0,0,1) -> (0,0,1,1) -> (0,0,1,1,1) -> (0,0,1,0)
  7     1     111  (0,0,1,0) -> (0,0,1,0,1) -> (0,0,1,0,1,1) -> (0,0,1,0,1,1,1)

Links

Crossrefs

Cf. A268289, A308551, A330261, A330265.

Programs

  • PARI
    See Links section.
Showing 1-3 of 3 results.

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