A269166 If A269162(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A269162(n)).

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 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, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0

Offset

0,26

Comments

a(n) gives the generational distance to the earliest finite ancestor when the binary expansion of n is interpreted as a pattern in Wolfram's Rule-30 cellular automaton or 0 if that pattern has no finite predecessors.

A110240 gives the record positions (after zero) and particularly, for n > 0, A110240(n) gives the first occurrence of n in this sequence.

See also comments in A269165.

Links

Antti Karttunen, Table of n, a(n) for n = 0..32767

Eric Weisstein's World of Mathematics, Rule 30

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

If A269162(n) = 0, then a(n) = 0, otherwise a(n) = 1 + a(A269162(n)).

Other identities. For all n >= 0:

a(A110240(n)) = n. [Works as a left inverse of sequence A110240.]

Prog

(Scheme)
;; This implementation is based on given recurrence and utilitizes the memoization-macro definec:
(definec (A269166 n) (let ((p (A269162 n))) (if (zero? p) 0 (+ 1 (A269166 p)))))
;; This one computes the same with tail-recursive iteration:
(define (A269166 n) (let loop ((n n) (p (A269162 n)) (s 0)) (if (zero? p) s (loop p (A269162 p) (+ 1 s)))))

Crossrefs

Cf. A110240, A269160, A269162.

Cf. A269164 (the indices of zeros after the initial zero).

Cf. A269165 (the earliest finite ancestor for n).

Cf. also A268389.

Sequence in context: A318141 A085979 A330986 * A377081 A331238 A330985

Adjacent sequences: A269163 A269164 A269165 * A269167 A269168 A269169

Keyword

nonn

Author

Antti Karttunen, Feb 21 2016

Status

approved