A322008 1/(1 - Integral_{x=0..1} x^(x^n) dx), rounded to the nearest integer.

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

Offset: 0

Author

M. F. Hasler, Mar 01 2019

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)

PARI

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

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)