A080924 Jacobsthal gap sequence.

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

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 sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

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: A141459 A318142 A176727 * A232179 A128042 A108083

Adjacent sequences: A080921 A080922 A080923 * A080925 A080926 A080927

Keyword

nonn,easy

Author

Paul Barry, Feb 26 2003

Status

approved