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%I #15 Feb 23 2016 12:08:58
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%H Antti Karttunen, <a href="/A269175/b269175.txt">Table of n, a(n) for n = 0..32768</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%o (Scheme)
%o (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.
%o ;; Somewhat optimized version:
%o (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)))))))))
%Y Cf. A000225, A000931, A182097, A269174.
%Y Cf. A269176 (indices of zeros), A269177 (of nonzeros), A269178 (of ones).
%K nonn
%O 0,32
%A _Antti Karttunen_, Feb 22 2016
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