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.

A019774 Decimal expansion of sqrt(e).

Original entry on oeis.org

1, 6, 4, 8, 7, 2, 1, 2, 7, 0, 7, 0, 0, 1, 2, 8, 1, 4, 6, 8, 4, 8, 6, 5, 0, 7, 8, 7, 8, 1, 4, 1, 6, 3, 5, 7, 1, 6, 5, 3, 7, 7, 6, 1, 0, 0, 7, 1, 0, 1, 4, 8, 0, 1, 1, 5, 7, 5, 0, 7, 9, 3, 1, 1, 6, 4, 0, 6, 6, 1, 0, 2, 1, 1, 9, 4, 2, 1, 5, 6, 0, 8, 6, 3, 2, 7, 7, 6, 5, 2, 0, 0, 5, 6, 3, 6, 6, 6, 4
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Also where x^(x^(-2)) is a maximum. - Robert G. Wilson v, Oct 22 2014
e^(1/2) maximizes the value of x^(c/(x^2)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

Examples

			1.6487212707001281468486507878141635716537761007101480115750...

Links

Crossrefs

Cf. A000354, A001113, A058281 for continued fraction for sqrt(e), A019775.

Programs

  • Maple
    evalf(sqrt(exp(1)), 120); # Muniru A Asiru, Aug 16 2018
  • Mathematica
    RealDigits[N[Sqrt[E],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)
  • PARI
    default(realprecision, 20080); x=sqrt(exp(1)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019774.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009

Formula

sqrt(e) = Sum_{n>=0} 1/(2^n*n!) = Sum_{n>=0} 1/(2n)!!. - Daniel Forgues, Apr 17 2011
sqrt(e) = 1 + Sum_{n>0} Product_{i=1..n} 1/(2n). - Ralf Stephan, Sep 11 2013
Continued fraction representation: sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). See A000354 for details. - Peter Bala, Jan 30 2015
sqrt(e) = (1/2)*( 1 + (3 + (5 + (7 + ...)/6)/4)/2 ) = 1 + (1 + (1 + (1 + ...)/6)/4)/2. - Rok Cestnik, Jan 19 2017
sqrt(e) = 2*Sum_{n >= 0} 1/((1 - 4*n^2)*(2^n)*n!). - Peter Bala, Jan 16 2022
sqrt(e) = (16/31)*(1 + Sum_{n>=1} (1/2)^n*((1/2)*n^3 + (1/2)*n + 1)/n!). - Alexander R. Povolotsky, Jul 01 2022
sqrt(e) = Sum_{n >= 0} (n + 1/2)/(2^n*n!). - Peter Bala, Jun 29 2024
Equals i^(-i/Pi), where i denotes the imaginary unit. - Stefano Spezia, Mar 01 2025

A066318 Number of necklaces with n labeled beads of 2 colors.

Original entry on oeis.org

2, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400, 163499212800, 3923981107200, 102023508787200, 2856658246041600, 85699747381248000, 2742391916199936000, 93241325150797824000, 3356687705428721664000, 127554132806291423232000
Offset: 1

Views

Author

Christian G. Bower, Dec 13 2001

Keywords

Comments

In the normal probability distribution with mean 0 and standard deviation 1, the expected value E[|x|^(2n-1)] = a(n)/sqrt(2*Pi), while E[|x|^(2n)] = E[x^(2n)] = A001147(n). - Stanislav Sykora, Jan 15 2017

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 66 (2.1.27,29).

Links

Crossrefs

Apart from initial term, same as A032184.
Cf. A000165, A000796, A001147, A019775, A019727.

Programs

  • GAP
    a_n:=List([1..10], n->Factorial(n-1)*2^n); # Stefano Spezia, Nov 17 2018
  • Magma
    [Factorial(n-1)*2^n: n in [1..20]]; // Vincenzo Librandi, Sep 23 2011
  • Maple
    with(combstruct):A:=[N,{N=Cycle(Union(Z$2))},labeled]: seq(count(A,size=n),n=1..18); # Zerinvary Lajos, Oct 07 2007
# alternative Maple program:
a:= n-> 2*doublefactorial(2*n-2):
seq(a(n), n=1..20);  # Alois P. Heinz, Jun 22 2017
  • Mathematica
    mx = 18; Rest[ Range[0, mx]! CoefficientList[ Series[ Log[1/(1 - 2 x)], {x, 0, mx}], x]] (* Robert G. Wilson v, Sep 22 2011 *)
Table[(n-1)!*2^n,{n,20}] (* Harvey P. Dale, Dec 15 2011 *)
  • Maxima
    a(n):=(n-1)!*2^n$ makelist(a(n), n, 1, 10);  /* Stefano Spezia, Nov 21 2018 */
  • PARI
    apply( A066318=n->(n-1)!<M. F. Hasler, Jan 15 2017
  • Python
    import math
for n in range(1,10): print(math.factorial(n-1)*2**n, end=', ') # Stefano Spezia, Nov 17 2018
  • Sage
    [2^n*factorial(n-1) for n in (1..20)] # G. C. Greubel, Nov 21 2018

Formula

a(n) = (n-1)!*2^n.
E.g.f.: log(1/(1-2*x)).
Let gd(x,n) = (d^n/dx^n)(exp(-(1/2)*x^2)*sqrt(2)/(2*sqrt(Pi))) = (-1)^((1/2)*n)*(x^2)^((1/2)*n)*2^(-(1/2)*n+1/2)*(exp(I*Pi*n)+1)/(4*sqrt(Pi)*GAMMA(1+(1/2)*n)) be the n-th derivative of the standard Gaussian distribution. Evaluating gd(x,n) at x=1 gives gd(1,n) = 2^(-(1/2)*n+1/2)*(exp(I*Pi*n)+1)*(-1)^((1/2)*n)/(4*sqrt(Pi)*GAMMA(1+(1/2)*n)). A066318 is the denominator of the even summands of the Taylor series expansion of the Gaussian distribution evaluated at x=1. a(n)=denom(gd(1, 2*n))/sqrt(Pi). - Stephen Crowley, May 16 2009
a(n) = 2*(n-1)*a(n-1). - R. J. Mathar, Sep 10 2012
G.f.: G(0), where G(k)= 1 + 1/(1 - 1/(1 + 1/(2*k+2)/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) = 2 * (2*n-2)!! = 2 * A000165(n-1). - Alois P. Heinz, Jun 22 2017
a(n) = (sqrt(Pi)/Gamma((2*n+3)/2))*Product_{k=0..n-1} binomial(2*(n-k)+1,2). - Stefano Spezia, Nov 17 2018
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=1} 1/a(n) = sqrt(e)/2 (A019775).
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/(2*sqrt(e)). (End)

A385496 Decimal expansion of 1 - exp(1/2)/2.

Original entry on oeis.org

1, 7, 5, 6, 3, 9, 3, 6, 4, 6, 4, 9, 9, 3, 5, 9, 2, 6, 5, 7, 5, 6, 7, 4, 6, 0, 6, 0, 9, 2, 9, 1, 8, 2, 1, 4, 1, 7, 3, 1, 1, 1, 9, 4, 9, 6, 4, 4, 9, 2, 5, 9, 9, 4, 2, 1, 2, 4, 6, 0, 3, 4, 4, 1, 7, 9, 6, 6, 9, 4, 8, 9, 4, 0, 2, 8, 9, 2, 1, 9, 5, 6, 8, 3, 6, 1, 1, 7, 3, 9, 9, 7, 1, 8, 1, 6, 6, 7, 8, 4, 9, 8, 5, 6, 6
Offset: 0

Views

Author

Artur Jasinski, Jun 30 2025

Keywords

Examples

			0.17563936464993592657567460609291821417311194964492...

Links

Crossrefs

Cf. A019775.

Programs

  • Mathematica
    RealDigits[1 - Exp[1/2]/2, 10, 105][[1]]

Formula

Equals 1 - A019775.
Equals Sum_{n>=1} 1/n! * Sum_{k>=2} log(k)^n/(k!*2^k).
Showing 1-3 of 3 results.

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

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