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.

A226572 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 12 2013

Keywords

Comments

Let h(x) be the greater of the two solutions of s + log(s) = x; then this sequence represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = LambertW(exp(2)) = 1.5571455989976... is the unique root of the equation s + log(s) = 2. Equation s - log(s) = 2 does have two roots, but they are s = -LambertW(-1,-exp(-2)) = 3.14619322062... (this sequence) and s = -LambertW(-exp(-2)) = 0.158594339563... (A202348, not A226571). - Vaclav Kotesovec, Jan 09 2014]
Apart from the first digit the same as A202321. - R. J. Mathar, Jun 15 2013

Examples

			2 + log 2 = 2.693147...
2 + log(2 + log 2) = 2.990710...
2 + log(2 + log(2 + log 2)) = 3.095510...
limit(f(n)) = 3.14619322062...

Links

Crossrefs

Cf. A202321, A202348, A226571, A226573, A226574.
Cf. also A052526.

Programs

  • Mathematica
    f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[2, 200]][[1]]  (* A226571 *)
d2 = RealDigits[g[2, 200]][[1]]  (* A226572 *)
s /. NSolve[s - Log[s] == 2, 200]  (* both constants *)
h[x_] := s /. NSolve[s - Log[s] == x]
Plot[h[x], {x, 1, 3}, PlotRange -> {0, 1}] (* bottom branch of h *)
Plot[h[x], {x, 1, 3}, PlotRange -> {1, 5}] (* top branch *)
  • PARI
    default(realprecision, 100); solve(x=3, 4, x - log(x) - 2) \\ Jianing Song, Dec 30 2018

Formula

Equals -LambertW(-1, -exp(-2)) = A202321 + 2. - Vaclav Kotesovec, Jan 09 2014

Extensions

Definition revised by N. J. A. Sloane, Dec 09 2017

A052525 Number of unlabeled rooted trees with n leaves in which the degrees of the root and all internal nodes are >= 3.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 10, 20, 36, 71, 136, 270, 531, 1070, 2147, 4367, 8895, 18262, 37588, 77795, 161444, 336383, 702732, 1472582, 3093151, 6513402, 13744384, 29063588, 61570853, 130669978, 277767990, 591373581, 1260855164
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Old name was "Non-planar unlabeled trees with neither unary nor binary nodes". I am leaving this alternative name here because it may help clarify the definitions of related sequences. - N. J. A. Sloane.

Examples

			For instance, with 7 leaves, the 6 choices are:
. [ *,*,*,*,*,*,* ]
. [ *,*,*,*,[ *,*,* ] ]
. [ *,*,*,[ *,*,*,* ] ]
. [ *,*,[ *,*,*,*,* ] ]
. [ *,*,[ *,*,[ *,*,* ] ] ]
. [ *,[ *,*,* ],[ *,*,* ] ]

Links

Crossrefs

Cf. A052524 and A052526.

Programs

  • Maple
    spec := [ S, {B=Union(S, Z), S=Set(B, 3 <= card)}, unlabeled ]: seq(combstruct[ count ](spec, size=n), n=0..50);

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.2318799173898687960533559522113115638..., c = 0.3390616344584879699709248904124... . - Vaclav Kotesovec, May 04 2015

Extensions

More terms from Paul Zimmermann, Oct 31 2002

A200318 E.g.f. satisfies: A(x) = x-1 + cosh(A(x)).

Original entry on oeis.org

1, 1, 3, 16, 120, 1156, 13608, 189316, 3039060, 55291336, 1124309208, 25268818576, 622008616320, 16642670404816, 480923246983728, 14926731083999296, 495243684302520000, 17491488288340789696, 655224017429959987968, 25947019896579324410176, 1083050878686674070676800
Offset: 1

Views

Author

Paul D. Hanna, Nov 15 2011

Keywords

Comments

a(n) is the number of leaf labeled rooted trees with n leaves in which the outdegrees of the root and all internal nodes are positive even integers. - Geoffrey Critzer, Jul 31 2016

Examples

			E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 120*x^5/5! +...
where A(1+x - cosh(x)) = x and A(x) = x-1 + cosh(A(x)).
The e.g.f. satisfies:
A(x) = x + (cosh(x)-1) + d/dx (cosh(x)-1)^2/2! + d^2/dx^2 (cosh(x)-1)^3/3! + d^3/dx^3 (cosh(x)-1)^4/4! +...
as well as the logarithmic series:
log(A(x)/x) = (cosh(x)-1)/x + d/dx (cosh(x)-1)^2/x/2! - d^2/dx^2 (cosh(x)-1)^3/x/3! + d^3/dx^3 (cosh(x)-1)^4/x/4! +...

Links

  • Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 128,(labeled hierarchies).

Crossrefs

Cf. A200317, A000311, A052526.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[1 + x - Cosh[x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(1+x-cosh(x+x^2*O(x^n))),n)}
for(n=1, 21, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, (cosh(x+x*O(x^n))-1)^m/x)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. satisfies:
(1) A(x) = Series_Reversion(1+x - cosh(x)).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (cosh(x) - 1)^n/x / n! ).
a(n) ~ n^(n-1) / (2^(1/4) * exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 10 2014
Showing 1-3 of 3 results.

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