A316253 - OEIS

A316253

Decimal expansion of the middle x such that 1/x + 1/(x+2) + 1/(x+3) = 3.

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

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OFFSET

1,2

COMMENTS

Equivalently, the least root of 3*x^3 + 12*x^2 + 8 x - 6;

Least: A316252;

Greatest root: A316254.

See A305328 for a guide to related sequences.

LINKS

Table of n, a(n) for n=1..86.

FORMULA

greatest root: -(4/3) + (4/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11))

***

middle: -(4/3) - (2/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11)) + 2 sqrt(2/3) sin((1/3) arctan(sqrt(391)/11))

****

least: -(4/3) - (2/3) sqrt(2) cos((1/3) arctan(sqrt(391)/11)) - 2 sqrt(2/3) sin((1/3) arctan(sqrt(391)/11))

EXAMPLE

greatest root: 0.4351172195495135109...

middle root: -1.650898528091803148...

least root: -2.784218691457710362...

MATHEMATICA

a = 1; b = 1; c = 1; u = 0; v = 2; w = 3; d = 3;

r[x_] := a/(x + u) + b/(x + v) + c/(x + w);

t = x /. ComplexExpand[Solve[r[x] == d, x]]

N[t, 20]

y = Re[N[t, 200]];

RealDigits[y[[1]]] (* A316254, greatest *)

RealDigits[y[[2]]] (* A316252, least *)

RealDigits[y[[3]]] (* A316253, middle *)

CROSSREFS

Cf. A305328, A316252, A316254.

Sequence in context: A096434 A193355 A248922 * A352527 A261992 A278760

Adjacent sequences: A316250 A316251 A316252 * A316254 A316255 A316256

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Sep 08 2018

STATUS

approved