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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. 34
 1, 7, 25, 111, 401, 1783, 6409, 28479, 102849, 456263, 1641433, 7287855, 26332369, 116815671, 420186569, 1865727615, 6741246849, 29904391303, 107568396185, 477630335215, 1725755276049, 7655529137527, 27537575631497 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See A245549 for binary equivalents. See A070952 for number of ON cells. - N. J. A. Sloane, Jul 28 2014 For n > 0: 3 < a(n+1) / a(n) < 5, floor(a(n+1)/a(n)) = A010702(n+1). - Reinhard Zumkeller, Jun 08 2013 Iterates of A269160 starting from a(0) = 1. See also A269168. - Antti Karttunen, Feb 20 2016 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 Erica Jen, Global properties of cellular automata, Journal of Statistical Physics 43 (1986), pp. 219-242. N. J. A. Sloane, Illustration of first 20 generations Eric Weisstein's World of Mathematics, Rule 30. Stephen Wolfram, Announcing the Rule 30 Prizes, 2019 FORMULA From Antti Karttunen, Feb 20 2016: (Start) a(0) = 1, for n >= 1, a(n) = A269160(a(n-1)). a(n) = A030101(A265281(n)). [The rule 30 is the mirror image of the rule 86.] A269166(a(n)) = n for all n >= 0. (End) From Antti Karttunen, Oct 05 2019: (Start) For n >= 1, a(n) = a(n-1) XOR 2*A328104(n-1). For n >= 1, a(n) = 2*a(n-1) XOR A327973(n). (End) EXAMPLE a(1)=1 because the automaton begins at first "generation" with one black cell: 1; a(2)=5 because one black cell, through Rule 30 at 2nd generation, produces three contiguous black cells: 111 (binary), so 7 (decimal); a(3)=25 because the third generation is "black black white white black" cells: 11001, so 25 (decimal). MATHEMATICA 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 *) PROG (Haskell) a110240 = foldl (\v d -> 2 * v + d) 0 . map toInteger . a070950_row -- Reinhard Zumkeller, Jun 08 2013 (Scheme, with memoization-macro definec) (definec (A110240 n) (if (zero? n) 1 (A269160 (A110240 (- n 1))))) ;; Antti Karttunen, Feb 20 2016 (PARI) A269160(n) = bitxor(n, bitor(2*n, 4*n)); A110240(n) = if(!n, 1, A269160(A110240(n-1))); \\ Antti Karttunen, Oct 05 2019 (Python) def A269160(n): return(n^((n<<1)|(n<<2))) def genA110240():     '''Yield successive terms of A110240 (Rule 30) starting from A110240(0)=1.'''     s = 1     while True:        yield s        s = A269160(s) def take(n, g):     '''Returns a list composed of the next n elements returned by generator g.'''     z = []     if 0 == n: return(z)     for x in g:         z.append(x)         if n > 1: n = n-1         else: return(z) take(30, genA110240()) # Antti Karttunen, Oct 05 2019 CROSSREFS This sequence, A070952, and A245549 all describe the same sequence of successive states. 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. Cf. A269165 (indices of ones in this sequence). Cf. A269166 (a left inverse). Left edge of A269168. Cf. also A265281, A269160, A328106. For bitwise XOR (and OR) combinations with other such 1D CA trajectories, see for example: A327971, A327972, A327973, A327976, A328103, A328104. Sequence in context: A200152 A255280 A327627 * A266810 A199893 A129791 Adjacent sequences:  A110237 A110238 A110239 * A110241 A110242 A110243 KEYWORD easy,nonn,base,changed AUTHOR Alexandre Wajnberg and Eric Angelini, Sep 06 2005 EXTENSIONS More terms from Eric W. Weisstein, Apr 08 2006 Offset corrected by Reinhard Zumkeller, Jun 08 2013 STATUS approved

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Last modified October 18 23:39 EDT 2019. Contains 328211 sequences. (Running on oeis4.)