A277450 - OEIS

%I #20 Apr 01 2017 17:27:30

%S 1,0,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,

%T 0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,

%U 0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1

%N a(1) = 1, a(n) = floor(n*Sum_{k=1..n-1} a(k)/2^k - Sum_{k=1..n-1} a(k)) for n > 1.

%C The mean density of ones in the sequence converges to a limit C, and the binary expansion of C is the sequence itself.

%C Let s(n) = Sum_{k=1..n} a(k), c(n) = Sum_{k=1..n} a(k)/2^k, and d(n) = s(n)/n.

%C By definition, s(n) = floor(n*c(n-1)) for n > 1.

%C We can show by induction that a(n) always equals 0 or 1.

%C Hence, c(n) converges to some limit C as n goes to infinity.

%C Also, d(n) = floor(n*c(n-1))/n ~ floor(n*C)/n ~ C as n goes to infinity (QED).

%C C = 0.58870955436366549427...

%C This constant C cannot be rational: the mean density of ones in the binary representation of a rational number u/v (with 0 < u < v and gcd(u,v) = 1) can be written as u'/v' with 0 < u' < v' and v' = A007733(v); as v' <= phi(v) < v, u/v never equals u'/v'.

%C Let b be the eventually periodic sequence with pre-period b(1)..b(14) = a(1)..a(14), and period 124 given by a(15)..a(138). The mean density of ones in b is the rational number 73/124 = 0.58870967... Of course, a and b cannot be equal, but a(n) = b(n) for all n < 65614. - _Michel Dekking_, Mar 30 2017

%t a[1] := 1; a[n_] := Floor[n*Sum[(a[k]/2^k), {k, 1, n - 1}]] - Sum[a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 10}] (* _G. C. Greubel_, Oct 17 2016 *)

%o (PARI) s=0; c=0; for (n=1, 100, a=if(n==1, 1, floor(n*c)-s); print1(a", "); s=s+a; c=c+a/2^n)

%Y Cf. A007733, A275973.

%K nonn,base,cons

%O 1

%A _Rémy Sigrist_, Oct 16 2016