A277450 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.

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, 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, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1

Offset

1

Comments

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

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.

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

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

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

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

C = 0.58870955436366549427...

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'.

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

Links

Table of n, a(n) for n=1..87.

Mathematica

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 *)

Prog

(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)

Crossrefs

Cf. A007733, A275973.

Sequence in context: A010059 A143580 A011749 * A188578 A322217 A104105

Adjacent sequences: A277447 A277448 A277449 * A277451 A277452 A277453

Keyword

nonn,base,cons

Author

Rémy Sigrist, Oct 16 2016

Status

approved