A269177 -id:A269177

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A269174

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

+10
13

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	LINKS	`Antti Karttunen, Table of n, a(n) for n = 0..32767` `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`
	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(2n) = 2a(n).` `a(n) = A057889(A161903(A057889(n))). [Rule 124 is the mirror image of rule 110.]`
	MATHEMATICA	`a[n_] := BitAnd[BitOr[n, 2n], BitOr[BitXor[n, 2n], BitXor[n, 4n]]];` `a /@ Range[0, 100] (* Jean-François Alcover, Feb 23 2020 *)`
	PROG	`(Scheme) (define (A269174 n) (A004198bi (A163617 n) (A003986bi (A048724 n) (A048725 n))))` `(Python) def a(n): return (n\|2n)&((n^(2n))\|(n^(4n))) # Indranil Ghosh, Apr 19 2017` `(Go)` `package main` `import "fmt"` `func main() {` `for n:=0; n<=100; n++{` `fmt.Println((n\|2n)&((n^(2n))\|(n^(4n))))}` `} // Indranil Ghosh, Apr 19 2017`
	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).`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Feb 22 2016`
	STATUS	`approved`

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.

+10
6

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,32`
	COMMENTS	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.
	LINKS	`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`
	PROG	`(Scheme)` `(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)))))))))`
	CROSSREFS	`Cf. A000225, A000931, A182097, A269174.` `Cf. A269176 (indices of zeros), A269177 (of nonzeros), A269178 (of ones).`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Feb 22 2016`
	STATUS	`approved`

A269176

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

+10
4

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`1,2`
	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	`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`
	PROG	`(Scheme, with Antti Karttunen's IntSeq-library)` `(define A269176 (ZERO-POS 1 1 A269175))`
	CROSSREFS	`Cf. A269174, A269175.` `Cf. A269177 (complement).`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Feb 22 2016`
	STATUS	`approved`

A269178

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

+10
4

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	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	`Antti Karttunen, Table of n, a(n) for n = 0..5141`
	PROG	`(Scheme, with Antti Karttunen's IntSeq-library)` `(define A269178 (ZERO-POS 0 0 (COMPOSE -1+ A269175)))`
	CROSSREFS	`Cf. A269174, A269175.` `Subsequence of A269177.`
	KEYWORD	`nonn`
	AUTHOR	`Antti Karttunen, Feb 22 2016`
	STATUS	`approved`

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