A364900 The n-volume of the unit regular n-simplex is sqrt(a(n))/A364901(n), with a(n) being squarefree.

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

Author

Jianing Song, Aug 12 2023

Keywords

• nonn
• easy

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.

Examples

			  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
		

Links

• Jianing Song, Table of n, a(n) for n = 0..10000
• Wikipedia, Simplex

Crossrefs

Cf. A007913, A364901, A120011, A020829, A364895, A001333.

Programs

PARI

a(n) = if(n%2, core((n+1)/2), core(n+1))

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.