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A257394 Decimal expansion of the negated x-coordinate of the inflection point of product{1 + x^k, k >= 1} that has minimal x-coordinate. 4
7, 8, 9, 8, 3, 4, 8, 5, 6, 9, 1, 0, 1, 1, 2, 2, 8, 5, 4, 4, 2, 6, 5, 1, 4, 6, 4, 7, 4, 6, 9, 3, 1, 2, 7, 1, 8, 4, 4, 4, 3, 4, 0, 5, 8, 4, 1, 8, 3, 1, 1, 6, 3, 8, 5, 3, 6, 6, 9, 3, 4, 6, 4, 7, 9, 5, 8, 9, 1, 4, 5, 6, 1, 4, 8, 1, 8, 0, 1, 6, 3, 5, 3, 3, 0, 9, 9, 2, 4, 4, 5, 6, 1, 2, 7, 6, 1, 2, 8, 5, 6, 8, 3, 9, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The function product{1 + x^k, k >= 1} has two inflection points:  (-0.78983..., 0.17671...) and (-0.23233..., 0.80084...).

LINKS

Table of n, a(n) for n=0..104.

EXAMPLE

x = -0.7898348569101122854426514647469312...

MATHEMATICA

f[x_] := f[x] = Product[(1 + x^k), {k, 1, 1000}];

p[x_, z_] := Sum[n/(x + x^(1 - n)), {n, z}]^2 + Sum[(n*x^(n - 2)*(n - x^n - 1))/(1 + x^n)^2, {n, z}];

Plot[f[x], {x, -1, 1}] (* plot showing 2 infl. pts. *)

t = x /. FindRoot[p[x, 1000], {x, -0.8}, WorkingPrecision -> 100] (* A257394 *)

u = f[t] (* A257395 *)

v = x /. FindRoot[p[x, 200], {x, -0.3}, WorkingPrecision -> 100]  (* A257396 *)

w = f[v] (* A257397 *)

RealDigits[t, 10][[1]]  (* A257394 *)

RealDigits[u, 10][[1]]  (* A257395 *)

RealDigits[v, 10][[1]]  (* A257396 *)

RealDigits[w, 10][[1]]  (* A257397 *)

(* Peter J. C. Moses, Apr 21 2015 *)

digits = 105; QP = QPochhammer; QPP[x_] := With[{dx = 10^-digits}, (QP[-1, x+dx] - QP[-1, x-dx])/(4*dx)]; x0 = x /. NMaximize[{QPP[x], -1 < x < -1/2}, x, WorkingPrecision -> 4 digits][[2]]; RealDigits[x0, 10, digits] // First (* Jean-François Alcover, Nov 19 2015 *)

CROSSREFS

Cf. A257395, A257396, A257397.

Sequence in context: A242022 A085676 A036793 * A196278 A006969 A195935

Adjacent sequences:  A257391 A257392 A257393 * A257395 A257396 A257397

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Apr 22 2015

EXTENSIONS

More digits from Jean-François Alcover, Nov 19 2015

STATUS

approved

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Last modified April 27 09:16 EDT 2017. Contains 285508 sequences.