OFFSET
1,2
COMMENTS
The radius of the inter-hypersphere of a regular simplex of dimension k with unit edge is given by r=Sqrt((k-j)/(2*(k+1)*(j+1))) where 0<=j<k and j is the dimension of the hypersurface that the inter-hypersphere is tangent to. j=0 gives the radius of the circum-hypersphere and j=k-1 gives the radius of the in-hypersphere. The sequence identifies those dimensions k for which at least one inter-hypersphere has a rational radius.
LINKS
Wikipedia, 8-simplex.
FORMULA
a(n)=k is the n-th occurrence for which k=sqrt((k-j)/(2*(k+1)*(j+1))) is rational for at least one value of j in the range 0<=j<k.
EXAMPLE
a(3)=8 because it is the 3rd value of k with k=8 that has an inter-hypersphere of rational radius. In fact in dimension 8 the regular simplex with unit edge has 4 inter-hyperspheres of rational radius. They are 2/3 at j=0 (circum-hypersphere), 1/3 at j=2, 1/6 at j=5 and 1/12 at j=7 (in-hypersphere).
MATHEMATICA
maxlen = 100; getlist[n_] := (d = 0; sublist = {}; While[d < n, If[IntegerQ[Numerator[Sqrt[(n-d)/(2(n+1)(d+1))]]*Denominator[Sqrt[(n-d)/(2(n+1)(d+1))]]], (sublist=Append[sublist, d]; d++), d++]]; sublist); m = 1; fulllist = {}; While[Length[fulllist] <= maxlen, If[Length[getlist[m]]==0, m++, (fulllist=Append[fulllist, m]; m++)]; ]; fulllist
CROSSREFS
KEYWORD
nonn
AUTHOR
Frank M Jackson, Feb 26 2013
STATUS
approved