A269177 -id:A269177 - 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

A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 0, 2, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 1, 1

Offset: 0

Antti Karttunen, Feb 22 2016

nonn

At positions A000225 seems to occur the record values of this sequence: 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, ... which seem to match with A000931 (Padovan sequence), or more exactly, with A182097 (Number of compositions (ordered partitions) into parts 2 and 3). Note that these values give also the number of predecessors for each "repunit-pattern" (2^n)-1 in Rule 110 cellular automaton, as rules 110 and 124 are mirror images of each other.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A000225, A000931, A182097, A269174.

Cf. A269176 (indices of zeros), A269177 (of nonzeros), A269178 (of ones).

(definec (A269175 n) (let loop ((p 0) (s 0)) (cond ((> p n) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0))))))) ;; Very straightforward and very slow.
;; Somewhat optimized version:
(definec (A269175 n) (if (zero? n) 1 (let ((nwid-1 (- (A000523 n) 1))) (let loop ((p (if (< n 2) 0 (A000079 nwid-1))) (s 0)) (cond ((> (A000523 p) nwid-1) s) (else (loop (+ 1 p) (+ s (if (= n (A269174 p)) 1 0)))))))))

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

Antti Karttunen, Feb 22 2016

nonn

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.

Antti Karttunen, Table of n, a(n) for n = 1..11672
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

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

Antti Karttunen, Feb 22 2016

nonn

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).

Antti Karttunen, Table of n, a(n) for n = 0..5141

Cf. A269174, A269175.

Subsequence of A269177.

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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A269175 a(n) = number of distinct k for which A269174(k) = n; number of finite predecessors for pattern encoded in the binary expansion of n in Wolfram's Rule 124 cellular automaton.

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

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

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