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.

Previous Showing 11-13 of 13 results.

A194625 Decimal expansion of the larger solution to x^x = 3/4.

Original entry on oeis.org

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

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Author

Jonathan Sondow, Sep 02 2011

Keywords

Comments

Since (1/e)^(1/e) < 3/4 < 1, the equation x^x = 3/4 has two solutions x = a and x = b with 0 < a < 1/e < b < 1. Both solutions are transcendental (see Proposition 2.2 in Sondow-Marques 2010).

Examples

			0.636262939294531019987513755204233173117867057936262294886540645406389214402799...

Links

Crossrefs

Cf. A030798 (x^x = 2), A072364 ((1/e)^(1/e)), A194624 (smaller solution to x^x = 3/4).

Programs

  • Mathematica
    x = x /. FindRoot[x^x == 3/4, {x, 0.7}, WorkingPrecision -> 120]; RealDigits[x, 10, 100] // First

A258707 Decimal expansion of ((1/exp(1))^(1/exp(1)))^2.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Jun 11 2015

Keywords

Examples

			.47914170878801531800402753289065712265788794309679401556441...

Links

Crossrefs

Cf. A072364.

Programs

  • Mathematica
    RealDigits[Exp[-1/Exp[1]]^2,10,104][[1]] (* Stefano Spezia, Jul 02 2025 *)
  • PARI
    exp(-1/exp(1))^2 \\ Michel Marcus, Sep 30 2017

Formula

Equals A072364^2.

A351917 G.f. A(x) = lim_{n->infinity} F(n), where F(0) = 1, F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1,2,3,...

Original entry on oeis.org

1, 1, 2, 7, 46, 428, 5297, 79803, 1425901, 29317826, 682661738, 17737351760, 508899803500, 15976473483553, 544826377810898, 20054787151757636, 792552513383685078, 33469556711571645662, 1504188914267750552104, 71681627511482293102424, 3610465782547332984625361
Offset: 0

Views

Author

Paul D. Hanna, Feb 25 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + 1425901*x^8 + 29317826*x^9 + 682661738*x^10 + ...
The g.f. A(x) equals the limit of the following process.
Start with F(0) = 1, then repeat F(n) = ( F(n-1)^n + (n*x)^n )^(1/n) for n = 1..N, for some positive integer N; the g.f. of this sequence equals the limit as N approaches infinity.
Example: start with F(0) = 1, then continue as follows.
F(1) = ( F(0)^1 + (1*x)^1 )^(1/1) = 1 + x;
F(2) = ( F(1)^2 + (2*x)^2 )^(1/2) = 1 + x + 2*x^2 - 2*x^3 + 4*x^5 - 6*x^6 - 2*x^7 + 22*x^8 + ...;
F(3) = ( F(2)^3 + (3*x)^3 )^(1/3) = 1 + x + 2*x^2 + 7*x^3 - 18*x^4 - 5*x^5 + 21*x^6 + 232*x^7 + ...;
F(4) = ( F(3)^4 + (4*x)^4 )^(1/4) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 - 197*x^5 + 21*x^6 - 216*x^7 + ...;
F(5) = ( F(4)^5 + (5*x)^5 )^(1/5) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 - 2479*x^6 + 1034*x^7 + ...;
F(6) = ( F(5)^6 + (6*x)^6 )^(1/6) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 - 37846*x^7 + ...;
F(7) = ( F(6)^7 + (7*x)^7 )^(1/7) = 1 + x + 2*x^2 + 7*x^3 + 46*x^4 + 428*x^5 + 5297*x^6 + 79803*x^7 + ...;
...
Continue in this way to obtain the g.f. A(x) as a limit of F(n) as n approaches infinity.
Related table.
The table of coefficients of x^k in A(x)^n, for k >= 0 and n >= 1, begins:
n=1: [1, 1,  2,   7,   46,  428,  5297,   79803, ...];
n=2: [1, 2,  5,  18,  110,  976, 11683,  172556, ...];
n=3: [1, 3,  9,  34,  198, 1677, 19388,  280326, ...];
n=4: [1, 4, 14,  56,  317, 2572, 28694,  405564, ...];
n=5: [1, 5, 20,  85,  475, 3711, 39945,  551180, ...];
n=6: [1, 6, 27, 122,  681, 5154, 53558,  720630, ...];
n=7: [1, 7, 35, 168,  945, 6972, 70035,  918016, ...]; ...
in which row n is comparable with F(n)^n given in the generating process shown above.
Related series.
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 155*x^4/4 + 1886*x^5/5 + 28908*x^6/6 + 517539*x^7/7 + 10708979*x^8/8 + 250010863*x^9/9 + 6513735398*x^10/10 + ...

Links

Programs

  • PARI
    {a(n) = my(A = 1 +x*O(x^n)) ; for(m=1,n, A = (A^m + (m*x)^m )^(1/m) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) ~ c * n^(n-1), where c = A072364 = exp(-exp(-1)). - Vaclav Kotesovec, Feb 27 2022
Previous Showing 11-13 of 13 results.

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