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 A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell. 1
 1, 4, 2, 121, 4, 2035, 8, 32743, 16, 524239, 32, 8388511, 64, 134217535, 128, 2147483263, 256, 34359737599, 512, 549755812351, 1024, 8796093019135, 2048, 140737488349183, 4096, 2251799813672959, 8192, 36028797018939391, 16384, 576460752303374335, 32768 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Rule 35 also generates this sequence. REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55. LINKS Robert Price, Table of n, a(n) for n = 0..999 Eric Weisstein's World of Mathematics, Elementary Cellular Automaton Index entries for linear recurrences with constant coefficients, signature (0,19,0,-50,0,32). FORMULA G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - Colin Barker, Dec 21 2015 a(n) = ((sqrt(2)+3)*((-1)^n+1)-6)*sqrt(2)^(n-3) - (2*4^n-1)*((-1)^n-1)/2. Therefore: a(n) = 2^(n/2) for even n; otherwise, a(n) = 2^(2*n+1)-3*2^((n-1)/2)-1. [Bruno Berselli, Dec 21 2015] EXAMPLE From Michael De Vlieger, Dec 21 2015: (Start) First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:               1                =               1 ->     1             1 0 0              =             100 ->     4           . . . 1 0            =              10 ->     2         1 1 1 1 0 0 1          =         1111001 ->   121       . . . . . . 1 0 0        =             100 ->     4     1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035   . . . . . . . . . 1 0 0 0    =            1000 ->     8 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743 (End) MATHEMATICA rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}][[k]], {rows-k+1, rows+k-1}], {k, 1, rows}][[k]], 2], {k, 1, rows}] Table[((Sqrt[2] + 3) ((-1)^n + 1) - 6) Sqrt[2]^(n - 3) - (2 4^n - 1) ((-1)^n - 1)/2, {n, 0, 30}] (* Bruno Berselli, Dec 22 2015 *) LinearRecurrence[{0, 19, 0, -50, 0, 32}, {1, 4, 2, 121, 4, 2035}, 40] (* Vincenzo Librandi, Dec 22 2015 *) PROG (PARI) Vec((1+4*x-17*x^2+45*x^3+16*x^4-64*x^5)/((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Dec 21 2015 (MAGMA) [IsEven(n) select 2^(n div 2) else 2^(2*n+1)-3*2^((n-1) div 2)-1: n in [0..30]]; // Bruno Berselli, Dec 22 2015 (Sage) [((sqrt(2)+3)*((-1)^n+1)-6)*sqrt(2)^(n-3)-(2*4^n-1)*((-1)^n-1)/2 for n in (0..30)] # Bruno Berselli, Dec 22 2015 CROSSREFS Cf. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074. Sequence in context: A236381 A201228 A010319 * A057167 A096683 A158903 Adjacent sequences:  A266066 A266067 A266068 * A266070 A266071 A266072 KEYWORD nonn,easy AUTHOR Robert Price, Dec 20 2015 STATUS approved

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Last modified November 16 23:51 EST 2018. Contains 317275 sequences. (Running on oeis4.)