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.

A051023 Middle column of rule-30 1-D cellular automaton, from a lone 1 cell.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1
Offset: 0

Views

Author

Eric W. Weisstein

Keywords

Comments

A092539(n) gives the value of prefix of length n+1, seen as a binary number. - Reinhard Zumkeller, Jun 08 2013
Also middle column of rule 86 1-D cellular automaton, from a lone 1 cell, as rule 86 is the mirror image of rule 30. - Antti Karttunen, Oct 03 2019

Links

Crossrefs

Cf. A070950, A269160 (rule 30), A071032, A269161 (rule 86).
Cf. A000035, A092539, A226474.
Cf. A327974 (adjacent bits xored), A327982 (partial sums), A327983 (run lengths).
Characteristic function of A327984 (gives the positions of ones in this sequence), A327985 (positions of zeros).
Cf. also A328100, A328101, A328102 (neighbor columns).
Cf. A365254 (converted to base 10).

Programs

  • Haskell
    a051023 n = a070950 n n  -- Reinhard Zumkeller, Jun 06 2013
  • Mathematica
    CellularAutomaton[30, {{1}, 0}, 101, {All, {0}}]//Flatten
  • PARI
    A051023(n) = ((A110240(n)>>n)%2);
\\ Or for fast creation of b-files:
A051023write(up_to) = { my(s=1, n=0); for(n=0,up_to, write("b051023.txt", n, " ", ((s>>n)%2)); s = A269160(s)); }; \\ Antti Karttunen, Oct 03 2019

Formula

a(n) = A070950(n,n). - Reinhard Zumkeller, Jun 06 2013
a(n) = 1 - A226474(n). - Reinhard Zumkeller, Jun 08 2013
From Antti Karttunen, Oct 04 2019: (Start)
a(n) = A000035(floor(A110240(n) / 2^n)).
For n>= 2, a(n) = (A328100(n) OR A328101(n)) XOR A328101(1+n). ["Sideways evaluation"]
(End)

Extensions

Corrected from 64th term by Daniel B. Cristofani (cristofd(AT)hevanet.com), Jan 07 2004

A327981 Distances between successive ones in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 3, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 4, 2, 2, 1, 1, 6, 3, 2, 1, 4, 1, 1, 4, 1, 2, 1, 2, 1, 2, 8, 4, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 6, 1, 3, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2019

Keywords

Comments

First differences of A327984, which gives indices of ones in A051023.

Examples

			The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
   0:              (1)
   1:             1(1)1
   2:            11(0)01
   3:           110(1)111
   4:          1100(1)0001
   5:         11011(1)10111
   6:        110010(0)001001
   7:       1101111(0)0111111
   8:      11001000(1)11000001
   9:     110111101(1)001000111
  10:    1100100001(0)1111011001
  11:   11011110011(0)10000101111
  12:  110010001110(0)110011010001
  13: 1101111011001(1)1011100110111
The distances between successive 1's in its central column (indicated here with parentheses) are 1-0 (as the first 1 is on row 0, and the second is on row 1), 3-1, 4-3, 5-4, 8-5, 9-8, 13-9, ..., that is, the first terms of this sequence: 1, 2, 1, 1, 3, 1, 4, ...

Links

Crossrefs

Cf. A051023, A110240, A269160, A327980, A327982, A327983, A327984.

Programs

  • Mathematica
    A327981list[upto_]:=Differences[Flatten[Position[CellularAutomaton[30,{{1},0},{upto,{{0}}}],1]]];A327981list[300] (* Paolo Xausa, Jun 27 2023 *)
  • PARI
    up_to = 105;
A269160(n) = bitxor(n, bitor(2*n, 4*n));
A327981list(up_to) = { my(v=vector(up_to), s=1, n=0, on=n, k=0); while(kA269160(s); if((s>>n)%2, k++; v[k] = (n-on); on=n)); (v); }
v327981 = A327981list(up_to);
A327981(n) = v327981[n];

Formula

a(n) = A327984(1+n) - A327984(n).

A327984 Positions of ones in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 9, 13, 15, 16, 19, 22, 23, 24, 26, 28, 29, 30, 33, 34, 35, 37, 39, 41, 42, 47, 48, 51, 53, 55, 56, 58, 60, 62, 63, 64, 65, 66, 67, 72, 73, 74, 75, 79, 81, 83, 84, 85, 91, 94, 96, 97, 101, 102, 103, 107, 108, 110, 111, 113, 114, 116, 124, 128, 129, 130, 131, 132, 134, 135, 136, 138, 141, 142, 143, 147
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2019

Keywords

Links

Crossrefs

Cf. A051023 (characteristic function), A110240, A269160.
Cf. A327981 (first differences), A327982 (a left inverse), A327985 (complement).

Programs

Formula

For all n >= 1, A327982(a(n)) = n.
Showing 1-3 of 3 results.

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