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A306617
Decimal expansion of a constant related to the asymptotics of A324425.
2
8, 2, 8, 8, 5, 9, 5, 7, 9, 6, 6, 9, 2, 7, 9, 2, 8, 6, 6, 9, 7, 2, 2, 9, 0, 2, 0, 7, 7, 5, 1, 0, 3, 0, 2, 6, 7, 6, 9, 1, 0, 5, 7, 5, 5, 9, 7, 7, 1, 2, 1, 1, 4, 5, 2, 4, 4, 0, 4, 0, 3, 3, 1, 7, 9, 5, 7, 1, 8, 3, 4, 3, 0, 2, 2, 1, 4, 7, 1, 8, 3, 7, 7, 6, 7, 1, 1, 3, 1, 1, 8, 9, 2, 7, 8, 7, 3, 0, 4, 0, 5, 4, 9, 3, 0, 9
OFFSET
0,1
COMMENTS
Ulrich Neumann found a closed form, see the "Mathematica Stack Exchange" link.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 [Terms beyond 79 were computed using Ulrich Neumann's program]
FORMULA
Equals limit_{n->infinity} (A324425(n)^(1/n^3))/n^2.
EXAMPLE
0.828859579669279286697229020775103026769105755977121145244040331795...
MAPLE
evalf(exp(integrate(log(x^2 + y^2 + z^2), x = 0..1, y = 0..1, z = 0..1)), 20);
evalf(exp(integrate(-2 + 2*sqrt(y^2 + z^2) * arctan(1/sqrt(y^2 + z^2)) + log(1 + y^2 + z^2), y = 0..1, z = 0..1)), 20);
MATHEMATICA
ixr = Exp[Integrate[1/3 (Log[1 + Sec[fi]^2] + (-7 + 3 Log[1 + Sec[fi]^2]) Sec[fi]^2 + 2 (Pi - 2 ArcTan[Sec[fi]]) Sec[fi]^3), {fi, 0, Pi/4}]]; Chop[N[ixr, 120]] (* A program by Ulrich Neumann added by Vaclav Kotesovec, Mar 03 2019. The calculation takes several minutes. *)
PROG
(PARI) exp(intnum(z=0, 1 , intnum(y=0, 1, intnum(x=0, 1, log(x^2 + y^2 + z^2)))))
(PARI) exp(intnum(z=0, 1 , intnum(y=0, 1, -2 + 2*sqrt(y^2 + z^2) * atan(1/sqrt(y^2 + z^2)) + log(1 + y^2 + z^2))))
CROSSREFS
Cf. A324425.
Sequence in context: A352473 A211269 A278809 * A278261 A296301 A019865
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Feb 28 2019
EXTENSIONS
More terms computed by Ulrich Neumann added by Vaclav Kotesovec, Mar 03 2019
STATUS
approved