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%I #19 Aug 20 2023 10:51:37
%S 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,
%T 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,
%U 3,55,7,57,29,59,30,61,31,7,2,65,33,67,34,69,35,71,1,73,37,3
%N The n-volume of the unit regular n-simplex is sqrt(a(n))/A364901(n), with a(n) being squarefree.
%C a(n) = 1 if and only if n = 2*k^2 - 1 or n = 4*k^2 - 4*k for k >= 1.
%C 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.
%H Jianing Song, <a href="/A364900/b364900.txt">Table of n, a(n) for n = 0..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Simplex">Simplex</a>
%F 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.
%e n | the n-volume of the
%e | unit regular n-simplex
%e 2 | sqrt(3)/4 = A120011
%e 3 | sqrt(2)/12 = A020829
%e 4 | sqrt(5)/96 = A364895
%e 5 | sqrt(3)/480
%e 6 | sqrt(7)/5760
%e 7 | 1/20160
%e 8 | 1/215040
%e 9 | sqrt(5)/5806080
%o (PARI) a(n) = if(n%2, core((n+1)/2), core(n+1))
%Y Cf. A007913, A364901, A120011, A020829, A364895, A001333.
%K nonn,easy
%O 0,3
%A _Jianing Song_, Aug 12 2023
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