This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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