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 A275973 A binary sequence due to Harold Jeffreys. 4
 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS Jeffreys defined this sequence in the context of sampling an events generator. Let a(n)=1 indicate that in the n-th sampling interval an event was detected; otherwise, set a(n)=0. This sequence's generator operates in such a way that a(1)=1 is followed by alternating blocks of 0's and blocks of 1's, each block having the same length as the whole sequence section which preceeds it. The pedagogical merit of the sequence consists in the fact that the would-be mean density of events, d(N) = (Sum_{n=1..N}a(n))/N = A275974(N)/N, does not converge to any limit when N grows to infinity. Rather, it oscillates (with exponentially growing cycle lengths) between liminf_{N->infinity} d(N) = 1/3 and limsup_{N->infinity} d(N) = 2/3. When interpreted as binary digits of a real number, the sequence evaluates to 1-A275975. In fact, it can be written as 1 - Sum_{k>=0}((-1)^k/2^2^k), with each pair of consecutive terms {1/2^2^(2m-1) - 1/2^2^(2m)}, for m = 1,2,3,..., giving rise to one of the blocks of one's. REFERENCES H. Jeffreys, Scientific Inference, Cambridge University Press, 3rd ed., 1973 (first published in 1931), Chapter III, page 47. LINKS Stanislav Sykora, Table of n, a(n) for n = 1..2100 FORMULA From Robert Israel, Aug 16 2016: (Start) G.f.: (1-x)*(1-x*Sum_{j>=0}(-1)^j*x^(2^j)). a(n) - a(n+1) = A154269(n). (End) a(1) = 1, a(2) = 0, for n > 2, a(n) = A030301(n-1) = A000035(A000523(n-1)). - Antti Karttunen, Sep 04 2016 MAPLE S:= series((1-x)^(-1)*(1-x*add((-1)^j*x^(2^j), j=0..9)), x, 1001): seq(coeff(S, x, j), j=1..1000); # Robert Israel, Aug 16 2016 PROG (PARI) \\ A vector-returning version adherent to the original definition: JeffreysSequence(nmax) = {  \\ Function returning a vector of length nmax   my(a=vector(nmax), n=0, p=1); a[n++]=1;   while(n

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Last modified May 28 17:57 EDT 2017. Contains 287241 sequences.