A120011 -id:A120011 - cp's OEIS frontend

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

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.

A179048 Decimal expansion of 25*sqrt(3)/4, the area of the equilateral triangle of side 5.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jun 26 2010

			10.825317547305483084546539634411702293392532836314878925348793621574581355680..

Cf. A002194, A010502, A120011, A179047.

a=b=c=5;area=Sqrt[(a+b-c)*(a-b+c)*(-a+b+c)*(a+b+c)]/4;RealDigits[N[area,200]]
RealDigits[(25*Sqrt[3])/4,10,120][[1]] (* Harvey P. Dale, May 30 2018 *)

25*sqrt(3)/4 \\ Charles R Greathouse IV, Jun 30 2011

Equals 25*A002194/4.

Offset corrected, keyword:cons inserted - R. J. Mathar, Jun 28 2010

A179275 Decimal expansion of 2*sqrt(Pi)/3^(1/4).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jul 07 2010

Also the side length of an equilateral triangle with area Pi (A000796), the area of a unit circle.

The area of an equilateral triangle with side length s is (sqrt(3)/4)s^2 = A120011*s^2, so A120011*(this constant)^2 = A000796.

			2.693547374177196721238160475092328667088670807308015892399206645495191607305...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A002161 (sqrt(Pi)), A011002 (3^1/4), A000796 (Pi), A002194 (sqrt(3)), A120011 (sqrt(3)/4).

RealDigits[2*Sqrt[Pi]/3^(1/4), 10, 100][[1]] (* G. C. Greubel, Mar 24 2017 *)

PARI
```
2*sqrt(Pi)/3^(1/4)
```

2*sqrt(Pi)/3^(1/4) = 2*A002161/A011002.

A194082 Sum{floor(sqrt(3)*k/2) : 1<=k<=n}.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 27, 34, 42, 51, 61, 72, 84, 96, 109, 123, 138, 154, 171, 189, 208, 227, 247, 268, 290, 313, 337, 362, 387, 413, 440, 468, 497, 527, 558, 590, 622, 655, 689, 724, 760, 797, 835, 873, 912, 952, 993, 1035, 1078, 1122, 1167, 1212

Offset: 1

Clark Kimberling, Aug 17 2011

nonn

Partial sums of A171970.
Comment from R. J. Mathar, Dec 02 2012 (Start):
a(n-1) is the number of unit squares regularly packed into the isosceles triangle of edge length n.
The triangle may be aligned with the Cartesian axes by putting its bottom edge on the horizontal axis, so its vertices are at (x,y) = (0,0), (n,0) and (n/2,sqrt(3)*n/2), see A010527.
The area inside the triangle is sqrt(3)*n^2/4 = A120011*n^2. There is an obvious upper limit of floor(sqrt(3)*n^2/4) = A171971(n) to the count of non-overlapping unit squares inside this triangle.
Regular packing: We place the first row of unit squares so they touch the bottom edge of the triangle. Their number is limited by the length of the horizontal section of the line y=1 inside the triangle, n-2*y/sqrt(3), which touches all of these first-row squares at their top.
The number of unit squares in the next row, between y=1 and y=2, is limited by the length of the horizontal section of the line y=2 inside the triangle, n-2*y/sqrt(3). Continuing, in row y=1, 2, ... we insert floor(n-2*y/sqrt(3)) unit squares, all with the same orientation.
The total number of squares is sum_{ y=1, 2, ..., floor(n*sqrt(3)/2) } floor( n-2*y/sqrt(3) ), and resummation yields, up to an index shift, this sequence here.
(End)

Cf. A171970.

r = Sqrt[3]/2;
c[k_] := Sum[Floor[j*r], {j, 1, k}];
Table[c[k], {k, 1, 90}]

a(n)=sum(k=1,n,sqrtint(3*k^2\4)) \\ Charles R Greathouse IV, Jan 06 2013

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Dec 08 2022

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))
  A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)
  A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...
where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/3))
  h = 1 - 1/2 = 0.5 (A020761)

			0.6141848493043784...

Index entries for transcendental numbers

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Maple
```
evalf(Pi/3-sqrt(3)/4);
```

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI
```
Pi/3 - sqrt(3)/4
```

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024

A387189 Decimal expansion of the smallest dihedral angle, in radians, in a pentagonal bipyramid (Johnson solid J_13).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 21 2025

This is the dihedral angle between triangular faces at the edge where the two pyramidal parts of the solid meet.

Also the dihedral angle between triangular faces in a pentagonal orthobicupola (Johnson solid J_30).

			1.3047162795687363719907812632876451487306158399...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Pentagonal bipyramid.
Wikipedia, Pentagonal orthobicupola.

Cf. A236367 (J_13 smallest dihedral angle).
Cf. other J_30 dihedral angles: A105199, A377995, A377996.
Cf. A179641 (J_13 volume), A120011 (J_13 surface area, divided by 10).
Cf. A384624 (J_30 volume), A384625 (J_30 surface area).
Cf. A010532, A386852.

First[RealDigits[ArcCos[(Sqrt[80] - 5)/15], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J13", "DihedralAngles"]], 10, 100]]

Equals arccos((4*sqrt(5) - 5)/15) = arccos((A010532 - 5)/15).

Equals 2*A386852.

A372047 Decimal expansion of sqrt(3)*Gamma(1/3)/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 22 2024

			1.160014413116984775294152416942826...

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p. 135, eq. (4.3.7).

Cf. A073005, A120011.

Maple
```
sqrt(3)/4*GAMMA(1/3) ; evalf(%) ;
```

RealDigits[Sqrt[3] * Gamma[1/3]/4, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

sqrt(3) * gamma(1/3) / 4 \\ Amiram Eldar, Jun 10 2024

Equals Integral_{x=0..oo} sin(x^3)/x^3 dx = A120011 * A073005.

A373732 a(n) = floor(4*n^2/sqrt(3)).

Original entry on oeis.org

0, 2, 9, 20, 36, 57, 83, 113, 147, 187, 230, 279, 332, 390, 452, 519, 591, 667, 748, 833, 923, 1018, 1117, 1221, 1330, 1443, 1561, 1683, 1810, 1942, 2078, 2219, 2364, 2514, 2669, 2829, 2992, 3161, 3334, 3512, 3695, 3882, 4073, 4270, 4471, 4676, 4886, 5101, 5320, 5544, 5773

Offset: 0

A. Timothy Royappa, Jun 17 2024

Maximum number of equilateral triangles with unit side, possibly cut into pieces, that can fit into a square of side n without overlapping.

The area of an equilateral triangle with unit side is sqrt(3)/4 (A120011), which gives the number a(n) of such triangles in a square of side n as at most floor(n^2/(sqrt(3)/4)).

			At most 9 unit equilateral triangles can fit into a square of side 2, so a(2) = 9.

Cf. A120011.

a(n) = floor(4*n^2/sqrt(3)).

A379339 Decimal expansion of Pi/8 + sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 21 2024

			0.82571178359094347818969200828640595226034748837448...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.3, p. 489.

Cf. A019675, A120011.

Mathematica
```
RealDigits[Pi/8+Sqrt[3]/4,10,100][[1]]
```

cp's OEIS Frontend

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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A179048 Decimal expansion of 25*sqrt(3)/4, the area of the equilateral triangle of side 5.

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A179275 Decimal expansion of 2*sqrt(Pi)/3^(1/4).

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A194082 Sum{floor(sqrt(3)*k/2) : 1<=k<=n}.

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A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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A387189 Decimal expansion of the smallest dihedral angle, in radians, in a pentagonal bipyramid (Johnson solid J_13).

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A372047 Decimal expansion of sqrt(3)*Gamma(1/3)/4.

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