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%I #7 Feb 27 2013 17:32:01
%S 1,7,8,17,23,31,35,49,55,62,71,74,80,89,95,97,119,120,127,143,146,161,
%T 174,199,215,223,224,241,244,251,287,288,293,299,305,323,337,359,360,
%U 362,383,391,424,440,449,479,483,489,503,511,566,574,575,577,587,599
%N Sequence of dimensions of a regular simplex with unit edge that has at least one inter-hypersphere with a rational radius (including the radii of the in-hypersphere and circum-hypersphere).
%C 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.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/8-simplex">8-simplex</a>.
%F 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.
%e 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).
%t 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
%Y Cf. A155946.
%K nonn
%O 1,2
%A _Frank M Jackson_, Feb 26 2013
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