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 A070950 Triangle read by rows giving successive states of cellular automaton generated by "Rule 30". 21
 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS If cell and right-hand neighbor are both 0 then new state of cell = state of left-hand neighbor; otherwise new state is complement of that of left-hand neighbor. A simple rule which produces apparently random behavior. "... probably the single most surprising discovery I have ever made" - Stephen Wolfram. Row n has length 2n+1. A110240(n) = A245549(n) = value of row n, seen as binary number. - Reinhard Zumkeller, Jun 08 2013 A070952 gives number of ON cells. - N. J. A. Sloane, Jul 28 2014 REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 27. LINKS Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened N. J. A. Sloane, Illustration of initial terms Eric Weisstein's World of Mathematics, Rule 30 S. Wolfram, A New Kind of Science FORMULA From Mats Granvik, Dec 06 2019: (Start) The following recurrence expresses the rules in rule 30, except that instead of If, Or, And, Not, we use addition, subtraction, and multiplication. T(n, 1) = 0 T(n, 2) = 0 T(1, 3) = 1 T(n, k) = [2*n + 1 >= k] ((1 - (T(n - 1, k - 2)*T(n - 1, k - 1)*T(n - 1, k)))*(1 - T(n - 1, k - 2)*T(n - 1, k - 1)*(1 - T(n - 1, k)))*(1 - (T(n - 1, k - 2)*(1 - T(n - 1, k - 1))*T(n - 1, k)))*(1 - ((1 - T(n - 1, k - 2))*(1 - T(n - 1, k - 1))*(1 - T(n - 1, k))))) + ((T(n - 1, k - 2)*(1 - T(n - 1, k - 1))*(1 - T(n - 1, k)))*((1 - T(n - 1, k - 2))*T(n - 1, k - 1)*T(n - 1, k))*((1 - T(n - 1, k - 2))*T(n - 1, k - 1)*(1 - T(n - 1, k)))*((1 - T(n - 1, k - 2))*(1 - T(n - 1, k - 1))*T(n - 1, k))). Discarding the term after the plus sign, multiplying/expanding the terms out and replacing all exponents with ones, gives us this simplified recurrence: T(n, 1) = 0 T(n, 2) = 0 T(1, 3) = 1 T(n, k) = T(-1 + n, -2 + k) + T(-1 + n, -1 + k) - 2 T(-1 + n, -2 + k) T(-1 + n, -1 + k) + (-1 + 2 T(-1 + n, -2 + k)) (-1 + T(-1 + n, -1 + k)) T(-1 + n, k). That in turn simplifies to: T(n, 1) = 0 T(n, 2) = 0 T(1, 3) = 1 T(n, k) = Mod(T(-1 + n, -2 + k) + T(-1 + n, -1 + k) + (1 + T(-1 + n, -1 + k)) T(-1 + n, k), 2). (End) EXAMPLE Triangle begins:   1;   1,1,1;   1,1,0,0,1;   1,1,0,1,1,1,1;   ... MATHEMATICA ArrayPlot[CellularAutomaton[30, {{1}, 0}, 50]] (* N. J. A. Sloane, Aug 11 2009 *) Clear[t, n, k]; nn = 10; t[1, k_] := t[1, k] = If[k == 3, 1, 0]; t[n_, k_] := t[n, k] = Mod[t[-1 + n, -2 + k] + t[-1 + n, -1 + k] + (1 + t[-1 + n, -1 + k]) t[-1 + n, k], 2]; Flatten[Table[Table[t[n, k], {k, 3, 2*n + 1}], {n, 1, nn}]] (*Mats Granvik, Dec 08 2019*) PROG (Haskell) a070950 n k = a070950_tabf !! n !! k a070950_row n = a070950_tabf !! n a070950_tabf = iterate rule30  where    rule30 row = f ([0, 0] ++ row ++ [0, 0]) where        f [_, _]          = []        f (u:ws@(0:0:_)) = u : f ws        f (u:ws)         = (1 - u) : f ws -- Reinhard Zumkeller, Feb 01 2013 CROSSREFS Cf. A070951, A070952 (row sums), A051023 (central terms). Cf. A071032 (mirror image, rule 86), A226463 (complemented, rule 135), A226464 (mirrored and complemented, rule 149). Cf. also A245549, A110240. Sequence in context: A194679 A111940 A129572 * A071031 A187037 A327866 Adjacent sequences:  A070947 A070948 A070949 * A070951 A070952 A070953 KEYWORD nonn,tabf,nice,easy AUTHOR N. J. A. Sloane, May 19 2002 EXTENSIONS More terms from Hans Havermann, May 24 2002 STATUS approved

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Last modified January 17 07:58 EST 2021. Contains 340214 sequences. (Running on oeis4.)