A036295 Numerator of Sum_{i=1..n} i/2^i.

0, 1, 1, 11, 13, 57, 15, 247, 251, 1013, 509, 4083, 4089, 16369, 2047, 65519, 65527, 262125, 131067, 1048555, 1048565, 4194281, 1048573, 16777191, 16777203, 67108837, 33554425, 268435427, 268435441, 1073741793, 67108863, 4294967263, 4294967279, 17179869149

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• frac

Comments

The fraction is twice the probability that the convex hull of n+2 points on a circle randomly chosen from a uniform distribution contains the center of the circle. This probability remains the same if the points are chosen from the circumference instead. - Lewis Chen, Jun 14 2025

References

• C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 95.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• A. F. Horadam, Oresme numbers, Fib. Quart., 12 (1974), 267-271.

Crossrefs

Cf. A036296 (denominators).

Cf. A000265, A000295, A006519.

Programs

Magma

[0] cat [Numerator(&+[i/2^i: i in [1..n]]): n in [1..40]]; // Vincenzo Librandi, Nov 09 2014

Maple

seq(numer(2-(n+2)/2^n), n=0..50); # Ridouane Oudra, Jul 16 2023

Mathematica

a[n_] := Module[{k, m}, For[k = 0; m = n + 2, EvenQ[m], k++, m/=2]; 2^(n + 1 - k) - m]
Table[Numerator[Sum[i/2^i, {i, n}]], {n, 40}] (* Alonso del Arte, Aug 12 2012 *)

PARI

concat(0, vector(100, n, numerator(sum(i=1, n, i/2^i)))) \\ Colin Barker, Nov 09 2014

PARI

a(n) = numerator(2-(n+2)/2^n); \\ Joerg Arndt, Jul 17 2023

Formula

a(n) = numerator(2-(n+2)/2^n).

If n+2=2^k*m with m odd, then a(n) = 2^(n+1-k) - m.

For n >= 1, a(n) = A000265(A000295(n+1)). - Peter Munn, May 30 2023

a(n) = A000295(n+1)/A006519(n+2). - Ridouane Oudra, Jul 16 2023

Numerators of coefficients in expansion of 2*x / ((1 - x) * (2 - x)^2). - Ilya Gutkovskiy, Aug 04 2023