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 A038183 One-dimensional cellular automaton 'sigma-minus' (Rule 90): 000,001,010,011,100,101,110,111 -> 0,1,0,1,1,0,1,0. 24
 1, 5, 17, 85, 257, 1285, 4369, 21845, 65537, 327685, 1114129, 5570645, 16843009, 84215045, 286331153, 1431655765, 4294967297, 21474836485, 73014444049, 365072220245, 1103806595329, 5519032976645, 18764712120593, 93823560602965 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Generation n (starting from the generation 0: 1) interpreted as a binary number. Observation: for n <= 15, a(n) = smallest number whose Euler totient is divisible by 4^n. This is not true for n = 16. - Arkadiusz Wesolowski, Jul 29 2012 Orbit of 1 under iteration of Rule 90 = A048725 = (n -> n XOR 4n). - M. F. Hasler, Oct 09 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig.9 Eric Weisstein's World of Mathematics, Rule 90 Wikipedia, Rule 90 Stephen Wolfram, Geometry of Binomial Coefficients, Amer. Math. Monthly, Volume 91, Number 9, November 1984, pages 566-571. S. Wolfram, O. Martin, and A.M. Odlyzko, Algebraic Properties of Cellular Automata (1984), Communications in Mathematical Physics, 93 (March 1984) 219-258. FORMULA a(n) = Product(((bit_n(n, i)*(2^(2^(i+1))))+1), i=0..inf); # A direct algebraic formula! a(n) = sum{k=0..n, (C(2*n, 2*k) mod 2)*4^(n-k)}. - Paul Barry, Jan 03 2005 a(2*n+1) = 5*a(2n); a(n+1) = a(n) XOR 4*a(n) where XOR is binary exclusive OR operator. - Philippe Deléham, Jun 18 2005 a(n) = A001317(2n). - Alex Ratushnyak, May 04 2012 EXAMPLE Successive states are: 1 101 10001 1010101 100000001 10100000101 1000100010001 101010101010101 10000000000000001 ... which when converted from binary to decimal give the sequence. - N. J. A. Sloane, Jul 21 2014 MAPLE bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2); # A recursive, cellular automaton rule version: sigmaminus := proc(n) option remember: if (0 = n) then (1) else sum('((bit_n(sigmaminus(n-1), i)+bit_n(sigmaminus(n-1), i-2)) mod 2)*(2^i)', 'i'=0..(2*n)) fi: end: MATHEMATICA r = 24; c = CellularAutomaton[90, {{1}, 0}, r - 1]; Table[FromDigits[c[[k, r - k + 1 ;; r + k - 1]], 2], {k, r}] (* Arkadiusz Wesolowski, Jun 09 2013 *) PROG (Python) a=1 for n in range(55): . print a, . a ^= a*4 # Alex Ratushnyak, May 04 2012 (PARI) vector(100, i, a=if(i>1, bitxor(a<<2, a), 1)) \\ M. F. Hasler, Oct 09 2017 CROSSREFS Cf. A006977, A006978, A038184, A038185 (other cellular automata), A000215 (Fermat numbers). Also alternate terms of A001317. Cf. A048710, A048720, A048757 (same 0/1-patterns interpreted in Fibonacci number system). Equals 4*A089893(n)+1. For right half of triangle see A245191. Cf. Sierpiński's gasket, A047999. Sequence in context: A149754 A149755 A002020 * A149756 A036756 A149757 Adjacent sequences:  A038180 A038181 A038182 * A038184 A038185 A038186 KEYWORD nonn AUTHOR Antti Karttunen, Feb 09 1999 STATUS approved

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