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A275973 A binary sequence due to Harold Jeffreys. 5

%I #36 Feb 18 2024 10:41:35

%S 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,

%T 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,

%U 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

%N A binary sequence due to Harold Jeffreys.

%C 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.

%C 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.

%C 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.

%D H. Jeffreys, Scientific Inference, Cambridge University Press, 3rd ed., 1973 (first published in 1931), Chapter III, page 47.

%H Alois P. Heinz, <a href="/A275973/b275973.txt">Table of n, a(n) for n = 1..16385</a> (first 2100 terms from Stanislav Sykora)

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F From _Robert Israel_, Aug 16 2016: (Start)

%F G.f.: (1-x)*(1-x*Sum_{j>=0}(-1)^j*x^(2^j)).

%F a(n) - a(n+1) = A154269(n). (End)

%F a(1) = 1, a(2) = 0, for n > 2, a(n) = A030301(n-1) = A000035(A000523(n-1)). - _Antti Karttunen_, Sep 04 2016

%p S:= series((1-x)^(-1)*(1-x*add((-1)^j*x^(2^j),j=0..9)), x, 1001):

%p seq(coeff(S,x,j),j=1..1000); # _Robert Israel_, Aug 16 2016

%p # secod Maple program:

%p b:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):

%p a:= proc(n) a(n):= `if`(n=1, 1, a(n-1)-b(n-1)) end:

%p seq(a(n), n=1..109); # _Alois P. Heinz_, Feb 18 2024

%o (PARI) \\ A vector-returning version adherent to the original definition:

%o JeffreysSequence(nmax) = { \\ Function returning a vector of length nmax

%o my(a=vector(nmax),n=0,p=1);a[n++]=1;

%o while(n<nmax,

%o for(k=2^(p-1)+1,2^p,a[n++]=0;if(n==nmax,break));

%o if(n<nmax,for(k=2^p+1,2^(p+1),a[n++]=1;if(n==nmax,break)));

%o p+=2;);

%o return(a);}

%o a = JeffreysSequence(2100) \\ An actual invocation

%o (PARI) \\ A function returning the n-th term:

%o a(n)={my(p=1,np=n-1);while(np,p++;np=np\2);return(bitand(p,1));}

%o (Scheme)

%o ;; A version after the above PARI-program. Here (A000035 n) = (modulo n 2) or (mod n 2), depending on the version of Scheme used:

%o (define (A275973_with_loop n) (let loop ((p 1) (np (- n 1))) (if (zero? np) (A000035 p) (loop (+ 1 p) (/ (- np (A000035 np)) 2)))))

%o ;; The above in turn reduces to this simple formula:

%o (define (A275973 n) (if (<= n 2) (A000035 n) (A030301 (- n 1))))

%o ;; _Antti Karttunen_, Sep 04 2016

%Y Cf. A000035, A000523, A030301, A275974 (partial sums), A275975, A154269.

%K nonn,base

%O 1

%A _Stanislav Sykora_, Aug 15 2016

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Last modified March 28 14:21 EDT 2024. Contains 371254 sequences. (Running on oeis4.)