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A275973
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A binary sequence due to Harold Jeffreys.
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5
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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
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OFFSET
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1
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COMMENTS
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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 precedes it.
The pedagogical merit of the sequence consists of 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.
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REFERENCES
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H. Jeffreys, Scientific Inference, Cambridge University Press, 3rd ed., 1973 (first published in 1931), Chapter III, page 47.
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LINKS
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FORMULA
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G.f.: (1-x)*(1-x*Sum_{j>=0}(-1)^j*x^(2^j)).
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MAPLE
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S:= series((1-x)^(-1)*(1-x*add((-1)^j*x^(2^j), j=0..9)), x, 1001):
# secod Maple program:
b:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):
a:= proc(n) a(n):= `if`(n=1, 1, a(n-1)-b(n-1)) end:
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PROG
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(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<nmax,
for(k=2^(p-1)+1, 2^p, a[n++]=0; if(n==nmax, break));
if(n<nmax, for(k=2^p+1, 2^(p+1), a[n++]=1; if(n==nmax, break)));
p+=2; );
return(a); }
a = JeffreysSequence(2100) \\ An actual invocation
(PARI) \\ A function returning the n-th term:
a(n)={my(p=1, np=n-1); while(np, p++; np=np\2); return(bitand(p, 1)); }
(Scheme)
;; A version after the above PARI-program. Here (A000035 n) = (modulo n 2) or (mod n 2), depending on the version of Scheme used:
(define (A275973_with_loop n) (let loop ((p 1) (np (- n 1))) (if (zero? np) (A000035 p) (loop (+ 1 p) (/ (- np (A000035 np)) 2)))))
;; The above in turn reduces to this simple formula:
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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