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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.

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