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A110240 Decimal form of binary integer produced by the ON cells at n-th generation following Wolfram's Rule 30 cellular automaton starting from a single ON-cell represented as 1.

%I A110240 #97 Feb 21 2025 16:43:59
%S A110240 1,7,25,111,401,1783,6409,28479,102849,456263,1641433,7287855,
%T A110240 26332369,116815671,420186569,1865727615,6741246849,29904391303,
%U A110240 107568396185,477630335215,1725755276049,7655529137527,27537575631497
%N A110240 Decimal form of binary integer produced by the ON cells at n-th generation following Wolfram's Rule 30 cellular automaton starting from a single ON-cell represented as 1.
%C A110240 See A245549 for binary equivalents. See A070952 for number of ON cells. - _N. J. A. Sloane_, Jul 28 2014
%C A110240 For n > 0: 3 < a(n+1) / a(n) < 5, floor(a(n+1)/a(n)) = A010702(n+1). - _Reinhard Zumkeller_, Jun 08 2013
%C A110240 Iterates of A269160 starting from a(0) = 1. See also A269168. - _Antti Karttunen_, Feb 20 2016
%C A110240 Also, the decimal representation of the n-th generation of the "Rule 66847740" 5-neighbors elementary cellular automaton starting with a single ON (black) cell. - _Philipp O. Tsvetkov_, Jul 17 2019
%H A110240 Reinhard Zumkeller, <a href="/A110240/b110240.txt">Table of n, a(n) for n = 0..1000</a>
%H A110240 Erica Jen, <a href="https://doi.org/10.1007/BF01010579">Global properties of cellular automata</a>, Journal of Statistical Physics 43 (1986), pp. 219-242.
%H A110240 Antti Karttunen, <a href="/A110240/a110240.png">Terms up to a(255) drawn as binary strings, with 1 bit = 3x3 pixels resolution</a>
%H A110240 Antti Karttunen, <a href="/A110240/a110240_1.png">Terms up to a(1023) drawn as binary strings, with 1 bit = 1 pixel resolution</a>
%H A110240 N. J. A. Sloane, <a href="/A070952/a070952.png">Illustration of first 20 generations</a>
%H A110240 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule30.html">Rule 30.</a>
%H A110240 Stephen Wolfram, <a href="https://blog.wolfram.com/2019/10/01/announcing-the-rule-30-prizes/">Announcing the Rule 30 Prizes</a>, 2019
%H A110240 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A110240 From _Antti Karttunen_, Feb 20 2016: (Start)
%F A110240 a(0) = 1, for n >= 1, a(n) = A269160(a(n-1)).
%F A110240 a(n) = A030101(A265281(n)). [The rule 30 is the mirror image of the rule 86.]
%F A110240 A269166(a(n)) = n for all n >= 0. (End)
%F A110240 From _Antti Karttunen_, Oct 05 2019: (Start)
%F A110240 For n >= 1, a(n) = a(n-1) XOR 2*A328104(n-1).
%F A110240 For n >= 1, a(n) = 2*a(n-1) XOR A327973(n). (End)
%e A110240 a(1)=1 because the automaton begins at first "generation" with one black cell: 1;
%e A110240 a(2)=5 because one black cell, through Rule 30 at 2nd generation, produces three contiguous black cells: 111 (binary), so 7 (decimal);
%e A110240 a(3)=25 because the third generation is "black black white white black" cells: 11001, so 25 (decimal).
%t A110240 rows = 23; ca = CellularAutomaton[30, {{1}, 0}, rows-1]; Table[ FromDigits[ ca[[k, rows-k+1 ;; rows+k-1]], 2], {k, 1, rows}] (* _Jean-François Alcover_, Jun 07 2012 *)
%o A110240 (Haskell)
%o A110240 a110240 = foldl (\v d -> 2 * v + d) 0 . map toInteger . a070950_row
%o A110240 -- _Reinhard Zumkeller_, Jun 08 2013
%o A110240 (Scheme)
%o A110240 ;; With memoization-macro definec.
%o A110240 (definec (A110240 n) (if (zero? n) 1 (A269160 (A110240 (- n 1)))))
%o A110240 ;; _Antti Karttunen_, Feb 20 2016
%o A110240 (PARI)
%o A110240 A269160(n) = bitxor(n, bitor(2*n, 4*n));
%o A110240 A110240(n) = if(!n,1,A269160(A110240(n-1))); \\ _Antti Karttunen_, Oct 05 2019
%o A110240 (Python)
%o A110240 def A269160(n): return(n^((n<<1)|(n<<2)))
%o A110240 def genA110240():
%o A110240     '''Yield successive terms of A110240 (Rule 30) starting from A110240(0)=1.'''
%o A110240     s = 1
%o A110240     while True:
%o A110240        yield s
%o A110240        s = A269160(s)
%o A110240 def take(n, g):
%o A110240     '''Returns a list composed of the next n elements returned by generator g.'''
%o A110240     z = []
%o A110240     if 0 == n: return(z)
%o A110240     for x in g:
%o A110240         z.append(x)
%o A110240         if n > 1: n = n-1
%o A110240         else: return(z)
%o A110240 take(30, genA110240())
%o A110240 # _Antti Karttunen_, Oct 05 2019
%Y A110240 Cf. A030101, A070950, A051023, A092539, A092540, A070952 (number of ON cells, the binary weight of terms), A100053, A100054, A100055, A094603, A094604, A000225, A074890, A010702, A245549, A269160, A269162.
%Y A110240 Cf. A269165 (indices of ones in this sequence).
%Y A110240 Cf. A269166 (a left inverse).
%Y A110240 Left edge of A269168.
%Y A110240 Cf. also A265281, A328106.
%Y A110240 For bitwise XOR (and OR) combinations with other such 1D CA trajectories, see for example: A327971, A327972, A327973, A327976, A328103, A328104.
%K A110240 easy,nonn,base
%O A110240 0,2
%A A110240 _Alexandre Wajnberg_ and _Eric Angelini_, Sep 06 2005
%E A110240 More terms from _Eric W. Weisstein_, Apr 08 2006
%E A110240 Offset corrected by _Reinhard Zumkeller_, Jun 08 2013