A364895 -id:A364895 - cp's OEIS frontend

A364896 Decimal expansion of the 4-volume of the unit regular 120-cell.

7, 8, 7, 8, 5, 6, 9, 8, 1, 0, 3, 4, 3, 3, 7, 9, 3, 3, 9, 9, 2, 1, 1, 6, 8, 5, 9, 1, 1, 3, 8, 8, 7, 4, 3, 6, 4, 9, 6, 4, 0, 8, 9, 8, 5, 8, 8, 1, 5, 3, 1, 4, 0, 8, 9, 0, 2, 7, 4, 5, 6, 3, 9, 5, 0, 3, 6, 0, 4, 3, 1, 3, 1, 4, 3, 6, 6, 3, 1, 1, 3, 5, 2, 1, 7, 9, 0, 5, 3, 9, 4, 7, 6, 7, 6, 0, 3, 7

Offset: 3

Jianing Song, Aug 12 2023

Decimal expansion of (1575+705*sqrt(5))/4.

			Equals 787.85698103433793399211...

Polytope Wiki, Hecatonicosachoron
Wikipedia, 120-cell

Decimal expansion of 4-volumes: A364895 (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), this sequence (120-cell), A364897 (600-cell).

Cf. A102769 (decimal expansion of the volume of the unit regular dodecahedron).

First[RealDigits[(1575 + 705*Sqrt[5])/4, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
(1575+705*sqrt(5))/4
```

A364897 Decimal expansion of the 4-volume of the unit regular 600-cell.

Original entry on oeis.org

2, 6, 4, 7, 5, 4, 2, 4, 8, 5, 9, 3, 7, 3, 6, 8, 5, 6, 0, 2, 5, 5, 7, 3, 3, 5, 4, 2, 9, 5, 7, 0, 4, 7, 6, 4, 7, 1, 5, 0, 3, 8, 6, 4, 7, 4, 7, 5, 7, 2, 0, 3, 5, 7, 7, 6, 6, 9, 3, 1, 0, 7, 7, 8, 3, 8, 1, 5, 7, 5, 5, 7, 8, 5, 2, 3, 6, 2, 8, 0, 6, 2, 1, 3, 4, 0, 0, 9, 0, 0, 5, 2, 3, 6, 7, 3, 8, 9, 2

Offset: 2

Jianing Song, Aug 12 2023

Decimal expansion of (50+25*sqrt(5))/4.

			Equals 26.47542485937368560255...

Wikipedia, 600-cell
Polytope Wiki, Hexacosichoron

Decimal expansion of 4-volumes: A364895 (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), A364896 (120-cell), this sequence (600-cell).

Cf. A102208 (decimal expansion of the volume of the unit regular icosahedron).

First[RealDigits[(50 + 25*Sqrt[5])/4, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
(50+25*sqrt(5))/4
```

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

Original entry on oeis.org

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

Jianing Song, Aug 12 2023

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.

			  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

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

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

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

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.

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

Original entry on oeis.org

1, 1, 4, 12, 96, 480, 5760, 20160, 215040, 5806080, 116121600, 1277337600, 30656102400, 398529331200, 11158821273600, 83691159552000, 5356234211328000, 30351993864192000, 3278015337332736000, 62282291409321984000, 2491291656372879360000, 52317124783830466560000

Offset: 0

Jianing Song, Aug 12 2023

			  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

Jianing Song, Table of n, a(n) for n = 0..425
Wikipedia, Simplex

Cf. A000188, A364900, A120011, A020829, A364895.

A000188(n) = sqrtint(n/core(n));
a(n) = n! * if(n%2, 2^((n-1)/2)/A000188((n+1)/2), 2^(n/2)/A000188(n+1))

The n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)), so a(n) = n! * 2^(n/2) / A000188(n+1) for even n and n! * 2^((n-1)/2) / A000188((n+1)/2) for odd n. It's easy to see that a(n) is an integer.

cp's OEIS Frontend

A364896 Decimal expansion of the 4-volume of the unit regular 120-cell.

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A364897 Decimal expansion of the 4-volume of the unit regular 600-cell.

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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.

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

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