A191871 a(n) = numerator(n^2 / 2^n).

0, 1, 1, 9, 1, 25, 9, 49, 1, 81, 25, 121, 9, 169, 49, 225, 1, 289, 81, 361, 25, 441, 121, 529, 9, 625, 169, 729, 49, 841, 225, 961, 1, 1089, 289, 1225, 81, 1369, 361, 1521, 25, 1681, 441, 1849, 121, 2025, 529, 2209, 9, 2401, 625, 2601, 169, 2809, 729, 3025

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Jun 18 2011

Keywords

• nonn
• frac
• easy
• mult

Comments

a(n+1) = largest odd divisor of A000290(n+1). - Jeremy Gardiner, Aug 25 2013

In binary, remove all trailing zeros, then square. - Ralf Stephan, Aug 26 2013

A fractal sequence. The odd-numbered elements give the odd squares A016754. If these elements are removed, the original sequence is recovered. - Jeremy Gardiner, Sep 14 2013

a(n+1) is the denominator of the population variance of the n-th row of Pascal's triangle. - Chai Wah Wu, Mar 25 2018

Multiplicative because A000265 is. - Andrew Howroyd, Jul 26 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Index entries for sequences related to binary expansion of n.

Crossrefs

Cf. A000265, A000290, A016754.

Programs

GAP

List([0..60],n->NumeratorRat(n^2/2^n)); # Muniru A Asiru, Mar 31 2018

Maple

[seq(numer(n^2/2^n), n=0..60)]; # Muniru A Asiru, Mar 31 2018

Mathematica

a[n_] := Numerator[n^2/2^n]; Table[a[n], {n, 0, 200, 2}]

PARI

a(n)=(n>>valuation(n,2))^2 \\ Charles R Greathouse IV & M. F. Hasler, Jun 19 2011

Python

from _future_ import division
def A191871(n):
    while not n % 2:
        n //= 2
    return n**2 # Chai Wah Wu, Mar 25 2018

Formula

a(n) = A000265(n^2) = A000265(n)^2. - M. F. Hasler, Jun 19 2011

Recurrence: a(2n) = a(n), a(2n+1) = (2n+1)^2. - Ralf Stephan, Aug 26 2013

From Amiram Eldar, Nov 28 2022: (Start)

Multiplicative with a(2^e) = 1, and a(p^e) = p^(2*e) if p > 2.

Sum_{k=1..n} a(k) ~ (4/21) * n^3. (End)

Dirichlet g.f.: zeta(s-2)*(2^s-4)/(2^s-1). - Amiram Eldar, Jan 04 2023