A269173 -id:A269173 - cp's OEIS frontend

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

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

Antti Karttunen, Feb 22 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32767
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

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).

package main
import "fmt"
func main() {
    for n:=0; n<=100; n++{
    fmt.Println((n|2*n)&((n^(2*n))|(n^(4*n))))}
} // Indranil Ghosh, Apr 19 2017

a[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 23 2020 *)

def a(n): return (n|2*n)&((n^(2*n))|(n^(4*n))) # Indranil Ghosh, Apr 19 2017

(define (A269174 n) (A004198bi (A163617 n) (A003986bi (A048724 n) (A048725 n))))

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

A352528 The binary expansion of a(n) is obtained by applying the elementary cellular automaton with rule (2*n) mod 256 to the binary expansion of n.

Original entry on oeis.org

0, 1, 1, 2, 0, 4, 2, 6, 2, 11, 5, 12, 5, 12, 4, 12, 0, 17, 9, 26, 4, 21, 14, 30, 2, 19, 3, 18, 9, 25, 8, 24, 0, 33, 17, 50, 0, 36, 19, 54, 2, 35, 21, 52, 7, 38, 21, 52, 8, 41, 9, 42, 28, 61, 31, 62, 6, 39, 7, 38, 19, 51, 17, 48, 0, 65, 33, 98, 0, 68, 34, 103

Offset: 0

Rémy Sigrist, Mar 19 2022

The binary digit of a(n) at place value 2^k is a function of the binary digits of n at place values 2^(k+2), 2^(k+1) and 2^k (and of (2*n) mod 256).

We use even elementary cellular automaton rules, so "000" will always evolve to "0", and the binary expansion of a(n) will have finitely many 1's and will be correctly defined.

			For n = 13:
- we apply rule 26,
- the binary expansion of 26 being "00011010", we apply the following evolutions:
      111 110 101 100 011 010 001 000
       0   0   0   1   1   0   1   0
- the binary expansion of 13 (with leading zeros) is   "...0001101",
- the binary digit of a(13) at place value 2^0 is 0 (from     "101"),
- the binary digit of a(13) at place value 2^1 is 0 (from    "110"),
- the binary digit of a(13) at place value 2^2 is 1 (from   "011"),
- the binary digit of a(13) at place value 2^3 is 1 (from  "001"),
- the other binary digits of a(13)            are 0 (from "000"),
- so the binary expansion of a(13) is "1100",
- so a(13) = 12.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 2^20 terms
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to binary expansion of n
Index entries for sequences related to cellular automata

Cf. A269160, A269161, A269173, A269174.

a(n) = { my (v=0, m=n); for (k=0, oo, if (m==0, return (v), bittest(2*n, m%8), v+=2^k); m\=2) }

cp's OEIS Frontend

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

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