cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

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

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

A215712 Numerator of sum(i=1..n, 3*i/4^i ).

Original entry on oeis.org

3, 9, 81, 21, 1359, 2727, 21837, 21843, 349515, 699045, 5592393, 2796201, 89478471, 178956963, 1431655749, 1431655761, 22906492227, 45812984481, 366503875905, 22906492245, 5864062014783, 11728124029599, 93824992236861, 93824992236879, 1501199875790139
Offset: 1

Views

Author

Alonso del Arte, Aug 21 2012

Keywords

Comments

The limit as n goes to infinity is 4/3.

Examples

			a(4) = 21 because 3/4 + 6/16 + 9/64 + 12/256 = 3/4 + 3/8 + 9/64 + 3/64 = 48/64 + 24/64 + 9/64 + 3/64 = 84/64 = 21/16.

References

  • Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 96.

Links

Crossrefs

Cf. A215713 for the denominators.
A036295/A036296 is the same with i/2^i instead of 3i/4^i.
Cf. A122553.

Programs

  • Magma
    [Numerator(&+[3*i/4^i: i in [1..n]]): n in [1..25]]; // Bruno Berselli, Sep 03 2012
  • Mathematica
    Table[Numerator[Sum[3i/4^i, {i, n}]], {n, 40}]

Extensions

a(17) corrected by Vincenzo Librandi, Sep 04 2012

A215713 Denominator of sum(i=1..n, 3*i/4^i).

Original entry on oeis.org

4, 8, 64, 16, 1024, 2048, 16384, 16384, 262144, 524288, 4194304, 2097152, 67108864, 134217728, 1073741824, 1073741824, 17179869184, 34359738368, 274877906944, 17179869184, 4398046511104, 8796093022208, 70368744177664, 70368744177664, 1125899906842624
Offset: 1

Views

Author

Alonso del Arte, Aug 21 2012

Keywords

Comments

The odd-indexed terms are the even-indexed powers of 4 (A013709).

Examples

			a(4) = 16 because 3/4 + 6/16 + 9/64 + 12/256 = 3/4 + 3/8 + 9/64 + 3/64 = 48/64 + 24/64 + 9/64 + 3/64 = 84/64 = 21/16.

References

  • Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 96.

Links

Crossrefs

Cf. A215712 for the numerators. A036295/A036296 is very similar but with i/2^i instead of 3i/4^i. Cf. also A122553.

Programs

  • Magma
    [Denominator(&+[3*i/4^i: i in [1..n]]): n in [1..25]]; // Bruno Berselli, Sep 03 2012
  • Mathematica
    Table[Denominator[Sum[3i/4^i, {i, n}]], {n, 40}]
  • PARI
    vector(100, n, denominator(sum(i=1, n, 3*i/4^i))) \\ Colin Barker, Nov 09 2014
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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