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 A080924 Jacobsthal gap sequence. 6
 0, 1, 3, 1, 15, 1, 63, 1, 255, 1, 1023, 1, 4095, 1, 16383, 1, 65535, 1, 262143, 1, 1048575, 1, 4194303, 1, 16777215, 1, 67108863, 1, 268435455, 1, 1073741823, 1, 4294967295, 1, 17179869183, 1, 68719476735, 1, 274877906943, 1, 1099511627775, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Inverse binomial transform of A080925 From Peter Bala, Dec 26 2012: (Start) Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/4) = 0.79761 68651 30459 16010 ... = 1/(1 + 1/(3 + 1/(1 + 1/(15 + 1/(1 + 1/(63 + 1/(1 + 1/(255 + ...)))))))). See A111317. (End) Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Apr 19 2017 REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015 Eric Weisstein's World of Mathematics, Elementary Cellular Automaton S. Wolfram, A New Kind of Science Wolfram Research, Wolfram Atlas of Simple Programs Index entries for linear recurrences with constant coefficients, signature (-1,4,4). FORMULA a(2n) = 3*A001045(2n) = 3*A002450(n) = 4^n-1, a(2n+1)=1. a(n) = (2^n-2*(-1)^n+(-2)^n)/2. G.f.: x*(1+4*x)/((1+x)*(1+2*x)*(1-2*x)). E.g.f.: (exp(2*x)-2*exp(-x)+exp(-2*x))/2. MATHEMATICA CoefficientList[Series[x (1 + 4 x) / ((1 + x) (1 + 2 x) (1 - 2 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 05 2013 *) LinearRecurrence[{-1, 4, 4}, {0, 1, 3}, 42] (* Jean-François Alcover, Sep 21 2017 *) CROSSREFS Cf. A001045, A002450, A080926, A080927, A111317. Sequence in context: A214073 A141459 A176727 * A232179 A128042 A108083 Adjacent sequences:  A080921 A080922 A080923 * A080925 A080926 A080927 KEYWORD nonn,easy AUTHOR Paul Barry, Feb 26 2003 STATUS approved

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