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A074890
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Decimal form of binary integers produced by a modified version of Wolfram's Rule 30 one-dimensional cellular automaton.
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4
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1, 3, 6, 13, 25, 55, 100, 222, 401, 891, 1602, 3559, 6428, 14258, 25647, 56936, 102860, 228154, 410339, 910998, 1645813, 3650437, 6565453, 14576121, 26332935, 58407052, 105047514, 233217299, 421327294, 934513441, 1680759539
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OFFSET
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0,2
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COMMENTS
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According to the common nomenclature (see A110240), this is actually a Rule 86 for all but the least-significant bit, and a Rule 252 for the least-significant bit, because the (fractional) bits right from the binary dot are never set. As a side effect, the first 11 terms--not the 12th--can be reproduced by a(n)=floor(A110240(n)/2^n). [From R. J. Mathar, Apr 29 2009]
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 27.
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LINKS
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FORMULA
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If b(n) is current binary digit, perform for each digit to get next integer in sequence: b(n) = (b(n)==0 && b(n+1)==0) ? b(n-1) : 1-b(n-1);//Wolfram's Rule 30*
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EXAMPLE
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a(4)=13 because a(3)=6=0110(binary) and applying Rule 30 to each digit [(b(n)==0 && b(n+1)==0) ? b(n-1) : 1-b(n-1)]
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PROG
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/**Java class to generate sequence */ public class r30seqA { static String zero="0", one="1"; public static void main(String[] args) { int base10 = 1; System.out.println(base10); for(int i=0; i<30; i++) System.out.println(base10 = base10Convert(applyR30(Integer.toBinaryString(base10)))); }
static String applyR30(String base2) { int a0, a1, a2, newDigit; StringBuffer newBase2 = new StringBuffer(); for(int i=-1; i<base2.length(); i++){ if(i<1) a0=0; else a0=base2.substring(i-1, i).equals(zero)?0:1; if(i<0) a1=0; else a1=base2.substring(i, i+1).equals(zero)?0:1; if(i==base2.length()-1) a2=0; else a2=base2.substring(i+1, i+2).equals(zero)?0:1; newDigit = a1==0&&a2==0?a0:1-a0; //Wolfram's Rule 30 newBase2.append(newDigit==0?zero:one); } return newBase2.toString(); }
static int base10Convert(String R30) { int newBase10 = 0; for(int i=0; i<R30.length(); i++) if(!R30.substring(i, i+1).equals(zero)) newBase10 += powerOf2(R30.length()-i-1); return newBase10; }
static int powerOf2(int power) { int p = 1; for(int i=0; i<power; i++) p*=2; return p; } }
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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Jason E Sackett (jason(AT)heavyion.com), Sep 13 2002
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EXTENSIONS
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STATUS
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approved
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