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

Author

Antti Karttunen, Feb 22 2016

Keywords

• nonn

Links

• 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

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

Go

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

Mathematica

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

Python

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

Scheme

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

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