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.

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A010722 Constant sequence: the all 6's sequence.

Original entry on oeis.org

6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Continued fraction expansion of 3+sqrt(10). - Bruno Berselli, Mar 15 2011
Decimal expansion of Sum_{n >= 0} n/binomial(2*n+1, n) = 2/3. - Bruno Berselli, Sep 14 2015
Decimal expansion of 2/3. - Franklin T. Adams-Watters, Feb 23 2019

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Links

Crossrefs

Cf. A145429: decimal expansion of Sum_{n >= 0} n/binomial(2*n, n).
Cf. A000012, A007395, A010701, A010709, A010716.
First differences of A008588.

Programs

Formula

G.f.: 6/(1-x). - Bruno Berselli, Mar 15 2011
E.g.f.: 6*e^x. - Vincenzo Librandi, Jan 27 2012
a(n) = floor(1/(-n + csc(1/n))). - Clark Kimberling, Mar 10 2020

A195686 a(n) = C(2*n,n) / gcd(n,C(2*n,n)).

Original entry on oeis.org

1, 2, 3, 20, 35, 252, 154, 3432, 6435, 48620, 92378, 705432, 676039, 10400600, 20058300, 10341168, 300540195, 2333606220, 1512522550, 35345263800, 6892326441, 179419291480, 1052049481860, 8233430727600, 2687300306925, 126410606437752, 247959266474052
Offset: 0

Views

Author

Peter Luschny, Oct 06 2011

Keywords

Links

Crossrefs

Cf. A093526, A093527, A145429.

Programs

  • Maple
    A195686  := n -> binomial(2*n,n)/igcd(n,binomial(2*n,n));
  • Mathematica
    a[n_] := Numerator[Binomial[2n,n]/n]; Join[{1}, Table[a[n], {n, 100}]] (* Enrique Pérez Herrero, Mar 26 2012 *)

Formula

A093526(n) = a(n+1)/(n+2).
a(n) = numerator(C(2n,n)/n). - Enrique Pérez Herrero, Mar 26 2012
Sum_{n>=0} A093527(n)/a(n+1) = Sum_{n>=1} n/binomial(2*n,n) = 2/3 + 2*Pi/(9*sqrt(3)) (A145429). - Amiram Eldar, Jan 26 2022
a(n) = numerator((n + 1)*binomial(2*n+1, n)/(n*(2*n + 1))) for n > 0. - Stefano Spezia, Aug 06 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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