OFFSET
1,3
LINKS
Lee David Chung Lin, Mean distance between 2 points in adjacent voxels (cubes), Mathematics Stack Exchange, 2018.
FORMULA
From Amiram Eldar, Mar 21 2026: (Start)
Equals (496 - 85*sqrt(2) + 36*sqrt(3) - 20*sqrt(5) - 81*sqrt(6) + 42*Pi - 1344*arcsin(1/5) + 1344*log(phi) - 252*log(2+sqrt(3)) + 735*log((1+sqrt(6))/sqrt(5)) - 168*arctan(sqrt(2/3)) - 105*log(1+sqrt(2)) + 168*log(sqrt(2)+sqrt(3)) + 42*log(2+sqrt(5)))/630, where phi is the golden ratio (A001622).
EXAMPLE
1.1673977869261100434588494044456138771...
MATHEMATICA
e[a_, b_, c_] := Module[{r = Sqrt[a^2 + b^2 + c^2], r1 = Sqrt[b^2 + c^2], r2 = Sqrt[c^2 + a^2], r3 = Sqrt[a^2 + b^2]},
2*r/15 - (7/45)*((r - r1)*(r1/a)^2 + (r - r2)*(r2/b)^2 + (r - r3)*(r3/c)^2) + (8/(315*(a*b*c)^2))*(a^7 + b^7 + c^7 -
r1^7 - r2^7 - r3^7 + r^7) + (1/(15*a*(b*c)^2))*(b^6*ArcSinh[a/b] +
c^6*ArcSinh[a/c] - r1^2*(r1^4 - 8*(b*c)^2)*ArcSinh[a/r1]) + (1/(15*b*(c*a)^2))*(c^6*ArcSinh[b/c] +
a^6*ArcSinh[b/a] - r2^2*(r2^4 - 8*(c*a)^2)*ArcSinh[b/r2]) + (1/(15*c*(a*b)^2))*(a^6*ArcSinh[c/a] +
b^6*ArcSinh[c/b] - r3^2*(r3^4 - 8*(a*b)^2)*ArcSinh[c/r3]) - (4/(15*a*b*c))*(a^4*ArcSin[b*c/(r2*r3)] +
b^4*ArcSin[c*a/(r3*r1)] + c^4*ArcSin[a*b/(r1*r2)])];
RealDigits[2*e[1, 1/2, 1/2] - e[1/2, 1/2, 1/2], 10, 120][[1]] (* Amiram Eldar, Jan 20 2026 *)
PROG
(PARI) my(phi = quadgen(5)); (496 - 85*sqrt(2) + 36*sqrt(3) - 20*sqrt(5) - 81*sqrt(6) + 42*Pi - 1344*asin(1/5) + 1344*log(phi) - 252*log(2+sqrt(3)) + 735*log((1+sqrt(6))/sqrt(5)) - 168*atan(sqrt(2/3)) - 105*log(1+sqrt(2)) + 168*log(sqrt(2)+sqrt(3)) + 42*log(2+sqrt(5)))/630 \\ Amiram Eldar, Mar 21 2026
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Hugo Pfoertner, Jan 19 2026
EXTENSIONS
More terms from Amiram Eldar, Jan 20 2026
STATUS
approved