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A322008 1/(1 - Integral_{x=0..1} x^(x^n) dx), rounded to the nearest integer. 3
2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 123, 146, 171, 198, 227, 258, 291, 326, 364, 403, 444, 487, 532, 579, 628, 679, 733, 788, 845, 904, 965, 1028, 1093, 1160, 1230, 1301, 1374, 1449, 1526, 1605, 1686, 1769, 1855, 1942, 2031, 2122, 2215, 2310, 2407, 2506, 2608 (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 maximal, for a given number n of x's, for F[n](x) := (...(x^x)^x....)^x = x^(x^(n-1)), which converges pointwise to x^0 = x for all x < 1, as n -> oo. The corresponding integrals therefore tend to 1 as n -> oo. This sequence is a convenient measure of the distance of these integrals from 1.

See A322009 for the minimal values of such integrals.

LINKS

Table of n, a(n) for n=0..50.

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

FORMULA

Conjectures from Colin Barker, Mar 07 2019: (Start)

G.f.: (2 + x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^9 + x^10 - x^11) / ((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x^4)).

a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10) for n>11.

(End)

EXAMPLE

For n=0, Integral_{x=0..1} x^(x^0) dx = Integral_{x=0..1} x^1 dx = 1/2, so a(0) = 1/(1 - 1/2) = 1 / 0.5 = 2.

For n=1, Integral_{x=0..1} x^(x^1) dx = Integral_{x=0..1} x^x dx = A083648 = 0.78343..., so a(1) = round( 1 / (1 - 0.78343...)) = round( 1 / 0.21656...) = 5.

MAPLE

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

seq(a(n), n=0..50);  # Alois P. Heinz, Mar 01 2019

MATHEMATICA

f[n_] := Round[1/(1 - NIntegrate[x^(x^n), {x, 0, 1}])]; Array[f, 51, 0] (* Robert G. Wilson v, Mar 01 2019 *)

PROG

(PARI) apply( A322008(n)=1\/intnum(x=0, 1, 1-x^x^n), [0..50])

CROSSREFS

Cf. A322009; A000081, A222379, A222380, A306679; A083648.

Sequence in context: A159547 A002522 A217990 * A300164 A248193 A069987

Adjacent sequences:  A322005 A322006 A322007 * A322009 A322010 A322011

KEYWORD

nonn

AUTHOR

M. F. Hasler, Mar 01 2019

STATUS

approved

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)