Shortest path sequence in d dimensions

[David Wilson]

Let d be the number of dimensions, and let a(d,n) be the length of
the shortest polygonal path in R^d from (0,0,..,0) to (n,n,...n)
whose vertices have integer coordinates and edges have integer
length.

Clearly, a(1, n) = n, since vector (n) has length 2n.

Empirically, a(2, n) appears to be the least even k >= sqrt(2)*n.

I also computed a(3, n), and it appears that a(3, n) is the least
k >= sqrt(3)*n with the same parity as n.

a(4,n) = 2n is easy, since vector (n, n, n, n) has integer length 2n.

It is looking as if a(d, n) is the smallest k >= sqrt(d)*n with
k == dn (mod 2).

- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
   - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"