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A364900
The n-volume of the unit regular n-simplex is sqrt(a(n))/A364901(n), with a(n) being squarefree.
2
1, 1, 3, 2, 5, 3, 7, 1, 1, 5, 11, 6, 13, 7, 15, 2, 17, 1, 19, 10, 21, 11, 23, 3, 1, 13, 3, 14, 29, 15, 31, 1, 33, 17, 35, 2, 37, 19, 39, 5, 41, 21, 43, 22, 5, 23, 47, 6, 1, 1, 51, 26, 53, 3, 55, 7, 57, 29, 59, 30, 61, 31, 7, 2, 65, 33, 67, 34, 69, 35, 71, 1, 73, 37, 3
OFFSET
0,3
COMMENTS
a(n) = 1 if and only if n = 2*k^2 - 1 or n = 4*k^2 - 4*k for k >= 1.
a(n) = a(n+1) = 1 if and only if n = A001333(k)^2 - 2 for even k and A001333(k)^2 - 1 for odd k.
LINKS
Wikipedia, Simplex
FORMULA
The n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)), so a(n) = A007913(n+1) for even n and A007913((n+1)/2) for odd n.
EXAMPLE
n | the n-volume of the
| unit regular n-simplex
2 | sqrt(3)/4 = A120011
3 | sqrt(2)/12 = A020829
4 | sqrt(5)/96 = A364895
5 | sqrt(3)/480
6 | sqrt(7)/5760
7 | 1/20160
8 | 1/215040
9 | sqrt(5)/5806080
PROG
(PARI) a(n) = if(n%2, core((n+1)/2), core(n+1))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Aug 12 2023
STATUS
approved