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A322009 1/(Integral_{x=0..1} x^(x^(x^n)) dx - 1/2), rounded to the nearest integer. 2
4, 14, 33, 64, 110, 174, 260, 369, 506, 672, 872, 1108, 1382, 1699, 2061, 2472, 2933, 3448, 4021, 4653, 5349, 6110, 6941, 7844, 8822, 9878, 11015, 12237, 13545, 14943, 16435, 18023, 19709, 21498, 23392, 25394, 27507, 29734, 32079, 34543, 37131, 39844, 42687, 45662, 48772 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Linked to the problem of sorting parenthesized expressions (x^x....^x) (cf. A000081 and A222379, A222380) according to the value of their integral from 0 to 1: This value is minimal, for a given number n of x's, for G[n](x) := x^((...(x^x)^x....)^x) = x^(x^(x^(n-2))), which converges pointwise to x^(x^0) = x^1 = x for all x in [0,1], as n -> oo. The corresponding integrals therefore tend to 1/2 as n -> oo. This sequence is a convenient measure of the distance of these integrals from 1/2.

See A322008 for the maximal values of such integrals.

LINKS

Table of n, a(n) for n=0..44.

Vladimir Reshetnikov, Integrals of power towers, on MathOverflow.net, Feb. 26, 2019.

EXAMPLE

For n=0, Integral_{x=0..1} x^(x^(x^0)) dx = Integral_{x=0..1} x^x dx = A083648 = 0.7834..., and 1/(0.7834... - 0.5) = 1 / 0.2834... = 3.528..., so a(0) = round(3.528...) = 4.

For n=1, Integral_{x=0..1} x^(x^(x^1)) dx = Integral_{x=0..1} x^(x^x) dx = 0.5731..., and 1/(0.5731... - 0.5) = 1 / 0.0731... = 13.67..., so a(1) = round(13.67...) = 14.

MAPLE

Digits:= 20:

a:= n-> round(evalf(1/(int(x^(x^(x^n)), x=0..1)-1/2))):

seq(a(n), n=0..44);  # Alois P. Heinz, Mar 01 2019

MATHEMATICA

f[n_] := Round[1/(NIntegrate[x^(x^(x^n)), {x, 0, 1}, WorkingPrecision -> 24] - 1/2)]; Array[f, 45, 0] (* Robert G. Wilson v, Mar 01 2019 *)

PROG

(PARI) A322009(n)=1\/intnum(x=0, 1, x^x^x^n-x)

CROSSREFS

Cf. A322008; A000081, A222379, A222380; A083648.

Sequence in context: A099590 A078432 A124786 * A228272 A011935 A266020

Adjacent sequences:  A322006 A322007 A322008 * A322010 A322011 A322012

KEYWORD

nonn

AUTHOR

M. F. Hasler, Mar 01 2019

STATUS

approved

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Last modified January 25 03:49 EST 2020. Contains 331241 sequences. (Running on oeis4.)