A182196 Decimal expansion of the larger root of x^sqrt(x+1) = sqrt(x+1)^x

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

Offset

1,1

Comments

The other root is the golden ratio phi = 1.618033988... (see A001622).

See also the graph with the two roots (second Mathematica program).

Links

Table of n, a(n) for n=1..120.

Example

3.6015212845193509063188978...

Maple

Digits:=200:fsolve(x^sqrt(x+1) - sqrt(x+1)^x=0, x, 3..5);

Mathematica

RealDigits[ FindRoot[x^Sqrt[x+1] == Sqrt[x+1]^x, {x, {3, 5} }, WorkingPrecision -> 105] [[1, 2] ]] [[2]]
(****graph****)
f[x_] := x^Sqrt[x+1] ; g[x_] := Sqrt[x+1]^x
Plot[{f[x], g[x]}, {x, 1, 4}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1, 4}, WorkingPrecision -> 110]

Prog

(PARI) solve(x=3, 4, t=sqrt(x+1); t^x-x^t) \\ Charles R Greathouse IV, Apr 18 2012

Crossrefs

Sequence in context: A104613 A113565 A178567 * A334477 A278488 A367325

Adjacent sequences: A182193 A182194 A182195 * A182197 A182198 A182199

Keyword

nonn,cons

Author

Michel Lagneau, Apr 18 2012

Status

approved