|
| |
|
|
A222624
|
|
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).
|
|
0
|
|
|
|
1, 7, 8, 17, 23, 31, 35, 49, 55, 62, 71, 74, 80, 89, 95, 97, 119, 120, 127, 143, 146, 161, 174, 199, 215, 223, 224, 241, 244, 251, 287, 288, 293, 299, 305, 323, 337, 359, 360, 362, 383, 391, 424, 440, 449, 479, 483, 489, 503, 511, 566, 574, 575, 577, 587, 599
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
