A269162 a(0) = 0, for n > 0, a(n) = the least (necessarily also unique) k such that A269160(k) = n, or 0 if no such k exists.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 6, 5, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 14, 13, 12, 11, 10, 0, 8, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 30, 29, 28, 27, 26, 0, 24, 23, 22, 21, 20, 0, 0, 25

Offset: 0

Author

Antti Karttunen, Feb 20 2016

Keywords

• nonn

Comments

If n > 0 and a(n) > 0 then a(n) is the unique finite predecessor of the configuration encoded in the binary representation of n (A007088) when Wolfram's Rule 30 cellular automaton is applied.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16387
• Eric Weisstein's World of Mathematics, Rule 30
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A110240, A269160, A269163, A269164 (indices of zeros), A269165, A269166.

Programs

Mathematica

(* empirical *) a[n_] := Module[{k}, For[k = Floor[n/7], k <= Ceiling[n/3], k++, If[BitXor[k, BitOr[2k, 4k]] == n, Return[k]]]; 0]; Table[a[n], {n, 0, 16387}] (* Jean-François Alcover, Feb 23 2016 *)

Scheme

(define (A269162 n) (let loop ((p 0)) (cond ((= n (A269160 p)) p) ((> p n) 0) (else (loop (+ 1 p)))))) ;; Very slow implementation.
(define (A269162 n) (if (zero? n) n (let ((nwid-2 (- (A000523 n) 2))) (let loop ((p (if (< n 4) 0 (A000079 nwid-2)))) (let ((k (A269160 p))) (cond ((= n k) p) ((> (A000523 p) nwid-2) 0) (else (loop (+ 1 p))))))))) ;; Somewhat optimized.

Formula

Other identities. For all n >= 0:

a(A269160(n)) = n. [This sequence works as a left inverse of A269160.]

a(A110240(n+1)) = A110240(n).