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.

A036296 Denominator of Sum_{i=1..n} i/2^i.

Original entry on oeis.org

1, 2, 1, 8, 8, 32, 8, 128, 128, 512, 256, 2048, 2048, 8192, 1024, 32768, 32768, 131072, 65536, 524288, 524288, 2097152, 524288, 8388608, 8388608, 33554432, 16777216, 134217728, 134217728, 536870912, 33554432, 2147483648, 2147483648, 8589934592, 4294967296
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Sum_{i>=0} i/2^i = 2. - Alonso del Arte, Aug 15 2012

Examples

			a(4) = 8 because 1/2 + 2/4 + 3/8 + 4/16 = 1/2 + 1/2 + 3/8 + 1/4 = 1 + 5/8 = 13/8.

References

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

Links

Crossrefs

Cf. A036295 (numerators).
Cf. A000079, A006519.

Programs

  • Magma
    [1] cat [Denominator(&+[i/2^i: i in [1..n]]): n in [1..40]]; // Vincenzo Librandi, Nov 09 2014
  • Maple
    seq(denom(2-(n+2)/2^n), n=0..50); # Ridouane Oudra, Jul 16 2023
  • Mathematica
    Table[Denominator[Sum[i/2^i, {i, n}]], {n, 40}] (* Alonso del Arte, Aug 08 2012 *)
  • PARI
    concat(1, vector(100, n, denominator(sum(i=1, n, i/2^i)))) \\ Colin Barker, Nov 09 2014
  • PARI
    a(n) = denominator(2-(n+2)/2^n); \\ Joerg Arndt, Jul 17 2023

Formula

a(n) = denominator(2-(n+2)/2^n). - Sean A. Irvine, Oct 25 2020
a(n) = A000079(n)/A006519(n+2), for n>=1. - Ridouane Oudra, Jul 16 2023
Denominators 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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