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 1-5 of 5 results.

A269177 Numbers that have a finite predecessor in Wolfram's Rule 124 cellular automaton; numbers n for which A269175(n) > 0.

Original entry on oeis.org

0, 3, 6, 7, 11, 12, 14, 15, 19, 22, 24, 27, 28, 30, 31, 35, 38, 44, 47, 48, 51, 54, 55, 56, 59, 60, 62, 63, 67, 70, 76, 79, 88, 91, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 115, 118, 119, 120, 123, 124, 126, 127, 131, 134, 140, 143, 152, 155, 158, 159, 176, 179, 182, 183, 187, 188, 190, 191, 192, 195
Offset: 0

Views

Author

Antti Karttunen, Feb 22 2016

Keywords

Comments

Sequence A269174 sorted into ascending order with duplicates removed.
The indexing starts from zero, because a(0) = 0 is a special case in this sequence. (Zero is the only number which is its own predecessor).

Links

Crossrefs

Cf. A269174, A269175.
Cf. A269176 (complement).
Cf. A269178 (a subsequence).

A269176 Numbers not present in A269174; indices of zeros in A269175.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 23, 25, 26, 29, 32, 33, 34, 36, 37, 39, 40, 41, 42, 43, 45, 46, 49, 50, 52, 53, 57, 58, 61, 64, 65, 66, 68, 69, 71, 72, 73, 74, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 92, 93, 97, 98, 100, 101, 104, 105, 106, 109, 113, 114, 116, 117, 121, 122, 125, 128, 129
Offset: 1

Views

Author

Antti Karttunen, Feb 22 2016

Keywords

Comments

Numbers n for which there is no any k such that A269174(k) = n.
These are binary representations (shown in decimal) of Garden of Eden patterns in Wolfram's Rule 124 cellular automaton if infinite predecessors are forbidden.

Links

Crossrefs

Cf. A269174, A269175.
Cf. A269177 (complement).

A269178 Numbers that have a unique finite predecessor in Wolfram's Rule 124 cellular automaton; numbers n for which A269175(n) = 1.

Original entry on oeis.org

0, 3, 6, 7, 11, 12, 14, 15, 19, 22, 24, 27, 28, 30, 35, 38, 44, 47, 48, 51, 54, 55, 56, 60, 67, 70, 76, 79, 88, 91, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 119, 120, 131, 134, 140, 143, 152, 155, 158, 159, 176, 179, 182, 183, 187, 188, 190, 191, 192, 195, 198, 199, 203, 204, 206, 207, 211, 214, 216, 219
Offset: 0

Views

Author

Antti Karttunen, Feb 22 2016

Keywords

Comments

The indexing starts from zero, because a(0) = 0 is a special case in this sequence. (Zero is the only number which is its own predecessor).

Links

Crossrefs

Cf. A269174, A269175.
Subsequence of A269177.

A269174 Formula for Wolfram's Rule 124 cellular automaton: a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 11, 24, 27, 30, 31, 28, 31, 22, 19, 48, 51, 54, 55, 60, 63, 62, 59, 56, 59, 62, 63, 44, 47, 38, 35, 96, 99, 102, 103, 108, 111, 110, 107, 120, 123, 126, 127, 124, 127, 118, 115, 112, 115, 118, 119, 124, 127, 126, 123, 88, 91, 94, 95, 76, 79, 70, 67, 192, 195, 198, 199, 204, 207, 206, 203, 216
Offset: 0

Views

Author

Antti Karttunen, Feb 22 2016

Keywords

Links

Crossrefs

Cf. A003986, A003987, A004198, A048724, A048725, A057889, A269173, A161903.
Cf. A269175.
Cf. A269176 (numbers not present in this sequence).
Cf. A269177 (same sequence sorted into ascending order, duplicates removed).
Cf. A269178 (numbers that occur only once).
Cf. A267357 (iterates from 1 onward).

Programs

Formula

a(n) = A163617(n) AND A269173(n).
a(n) = A163617(n) AND (A048724(n) OR A048725(n)).
a(n) = (n OR 2n) AND ((n XOR 2n) OR (n XOR 4n)).
Other identities. For all n >= 0:
a(2*n) = 2*a(n).
a(n) = A057889(A161903(A057889(n))). [Rule 124 is the mirror image of rule 110.]
G.f.: (-3*x^3 - 2*x^2 - 3*x)/(x^4 - 1) + Sum_{k>=1}((2^(k + 1)*x^(2^k) - 2^(k + 1)*x^(14*2^(k - 2)))/((x^(2^(k + 2)) - 1)*(x - 1))). - Miles Wilson, Jan 25 2025

A161903 Convert n into a sequence of binary digits, apply one step of the rule 110 cellular automaton, and interpret the results as a binary integer.

Original entry on oeis.org

0, 3, 6, 7, 12, 15, 14, 13, 24, 27, 30, 31, 28, 31, 26, 25, 48, 51, 54, 55, 60, 63, 62, 61, 56, 59, 62, 63, 52, 55, 50, 49, 96, 99, 102, 103, 108, 111, 110, 109, 120, 123, 126, 127, 124, 127, 122, 121, 112, 115, 118, 119, 124, 127, 126, 125, 104, 107, 110, 111, 100, 103, 98, 97, 192, 195, 198, 199, 204, 207, 206, 205, 216, 219, 222, 223, 220, 223, 218, 217, 240, 243, 246, 247, 252, 255, 254, 253, 248, 251, 254, 255, 244, 247, 242, 241, 224, 227, 230, 231, 236
Offset: 0

Views

Author

Ben Branman, Jan 30 2011

Keywords

Comments

a(a(a(...1))) (n times) gives A006978(n)

Examples

			For n=19, the evolution after one step is
0, 1, 0, 0, 1, 1  (n=19)
1, 1, 0, 1, 1, 1  (a(n)=55)
So a(n)=55.

Links

Crossrefs

Cf. A006978, A070887, A071049, A180001, A186083, A204371.
Cf. also A269160, A269174, A269175.

Programs

  • Mathematica
    a[n_] :=
FromDigits[
  Drop[Part[CellularAutomaton[110, {IntegerDigits[n, 2], 0}], 1], -1],
   2];Table[a[n],{n,0,100}]

Formula

a(n) = A057889(A269174(A057889(n))). - Antti Karttunen, Jun 02 2018
Showing 1-5 of 5 results.

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