A020829 -id:A020829 - cp's OEIS frontend

A386411 Decimal expansion of the volume of an augmented truncated tetrahedron with unit edge.

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

Offset: 1

Paolo Xausa, Jul 21 2025

The augmented truncated tetrahedron is Johnson solid J_65.

			3.889087296526011384204643991576669716066597657...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented truncated tetrahedron.

Cf. A386412 (surface area).

Cf. A010466, A020829, A377275.

First[RealDigits[11/Sqrt[8], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J65", "Volume"], 10, 100]]

Equals 11/(2*sqrt(2)) = 11/A010466.
Equals A377275 + 10*A020829.
Equals the largest root of 8*x^2 - 121.

A041127 Denominators of continued fraction convergents to sqrt(72).

Original entry on oeis.org

1, 2, 33, 68, 1121, 2310, 38081, 78472, 1293633, 2665738, 43945441, 90556620, 1492851361, 3076259342, 50713000833, 104502261008, 1722749176961, 3550000614930, 58522759015841, 120595518646612, 1988051057361633, 4096697633369878, 67535213191279681

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A041126, A040063, A020829, A010524.

I:=[1, 2, 33, 68]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

Denominator/@Convergents[Sqrt[72], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[(1 + 2 x - x^2)/(x^4 - 34 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
a0[n_] := ((3+2*Sqrt[2])/(17+12*Sqrt[2])^n+(3-2*Sqrt[2])*(17+ 12*Sqrt[2])^n)/6 // Simplify
a1[n_] := (-1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/(12*Sqrt[2]) // FullSimplify
Flatten[MapIndexed[{a0[#],a1[#]}&,Range[20]]] (* Gerry Martens, Jul 10 2015 *)

a(n)=my(v=contfrac(sqrt(72),n),s=v[n]);forstep(k=n-1,1,-1,s=v[k]+1/s);denominator(s) \\ Charles R Greathouse IV, Jul 05 2011

G.f.: -(x^2-2*x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - Colin Barker, Nov 13 2013
a(n) = 34*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 11 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((3+2*sqrt(2))/(17+12*sqrt(2))^n+(3-2*sqrt(2))*(17+12*sqrt(2))^n)/6.
a1(n) = (-1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/(12*sqrt(2)). (End)

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.

A093525 Decimal expansion of 13/720 - Pi^2/15015.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Mar 30 2004

Average volume of a tetrahedron picked at random in a tetrahedron with unit volume.
Buchta & Reitzner announced this result in 1992, and Mannion (independently) proved it in 1994. Buchta & Reitzner proved a more general result in 2001. - Charles R Greathouse IV, Sep 04 2015
Klee (1969) conjectured that the average volume is 1/60 and stated that according to Monte Carlo experiments 1/57 is the integer-reciprocal closest to this value. - Amiram Eldar, Apr 09 2022

			0.0173982392...

Christian Buchta and Matthias Reitzner, What is the expected volume of a tetrahedron whose vertices are chosen at random from a given tetrahedron, Österreichische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, Vol. 129 (1992), pp. 63-68.
Christian Buchta and Matthias Reitzner, The convex hull of random points in a tetrahedron: Solution of Blaschke's problem and more general results, J. reine angew. Math., Vol. 536 (2001), pp. 1-29.
Victor Klee, What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body?, The American Mathematical Monthly, Vol. 76, No. 3 (1969), pp. 286-288.
David Mannion, The volume of a tetrahedron whose vertices are chosen at random in the interior of a parent tetrahedron, Advances in Applied Probability, Vol. 26, No. 3 (Sep 1994), pp. 577-596.
Eric Weisstein's World of Mathematics, Tetrahedron Tetrahedron Picking.
Index entries for transcendental numbers

Cf. A020829, A093524, A093591.

Join[{0}, RealDigits[13/720 - Pi^2/15015, 10, 100][[1]]] (* Amiram Eldar, Apr 09 2022 *)

13/720 - Pi^2/15015 \\ Charles R Greathouse IV, Sep 04 2015

Added initial 0 to make offset correct. - N. J. A. Sloane, Feb 08 2015

A219965 Number of non-intersecting unit cubes regularly packed into the tetrahedron of edge length n.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 19, 30, 45, 66, 94, 130, 172, 221, 278, 344, 422, 511, 611, 723, 848, 987, 1140, 1308, 1491, 1691, 1909, 2146, 2401, 2673, 2965, 3278, 3614, 3974, 4355, 4759, 5186, 5638, 6117, 6623, 7156, 7716, 8305, 8923, 9571, 10249, 10958, 11700, 12475, 13285, 14127, 15003, 15914, 16862, 17849, 18874, 19937, 21037, 22177, 23358, 24581, 25846, 27153, 28504

Offset: 1

R. J. Mathar, Dec 02 2012

nonn

The tetrahedron may be aligned with the Cartesian axes by putting its triangular basis on the horizontal plane, with four vertices at (x, y, z) = (0, 0, 0), (n, 0, 0),  (n/2, sqrt(3)*n/2, 0) and (n/2, n/(2*sqrt(3)), n*sqrt(2/3)) see A194082, A020769, A157697.
The volume of tetrahedron is a third times the area of the base triangle times height, (1/3) * (sqrt(3)*n^2/4) * n*sqrt(2/3) = n^3/(3*2^(3/2)) = A020829*n^3. This defines an obvious upper limit of floor(n^3/sqrt(72)) = A171973(n) of placing unit cubes into this tetrahedron.
Regular packing: We place the first layer of unit cubes so they touch the floor of the tetrahedron. Their number is limited by the area of the triangular horizontal section of the plane z=1 inside the tetrahedron, which touches all of them; this isosceles horizontal triangle has edge length E(n,z) = n-z*sqrt(3/2). This edge length is a linear interpolation for triangular horizontal cuts between z=0 at the bottom and the summit of the tetrahedron at z=n*sqrt(2/3).
This first layer confined by a triangle characterized by E(n,z) may host RegSquInTri(E) := sum_{y=1..floor(E*sqrt(3)/2)} floor(E-y*2/sqrt(3)) cubes, following recursively the same regular placement and counting strategy as for squares in isosceles triangles, see A194082.
The number of unit cubes in the next layer, between z=1 and z=2, is limited by the area of the horizontal section of the triangle z=2 inside the tetrahedron, where the triangle has edge length n-z*sqrt(3/2).
So in layer z=1, 2, ... we insert ReqSquInTri(E(n,z)) cubes. a(n) is the sum over all these layers with z limited by the z-value of the vertex at the summit.
There is a generalization to placing unit cubes of higher dimensions into higher dimensional tetrahedra.
The growth is expected to be roughly equal to the growth of A000292.

R. J. Mathar, Illustration of 30 cubes at n=8
R. J. Mathar, Illustration of 66 cubes at n=10
R. J. Mathar, Illustration of 221 cubes at n=14

# Number or squares in isosceles triangle of edge length n.
RegSquInTri := proc(n)
        add(floor(n-2*y/sqrt(3)), y=1..floor(n*sqrt(3)/2)) ;
end proc:
A219965 := proc(n)
        local a,z,triedg ;
        a := 0 ;
        for z from 1 to floor(n*sqrt(2/3)) do
                triedg := n-z*sqrt(3/2) ;
                a := a+ RegSquInTri(triedg) ;
        end do:
        return a;
end proc:

a(n) <= A171973(n).

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A386411 Decimal expansion of the volume of an augmented truncated tetrahedron with unit edge.

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A041127 Denominators of continued fraction convergents to sqrt(72).

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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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A093525 Decimal expansion of 13/720 - Pi^2/15015.

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A219965 Number of non-intersecting unit cubes regularly packed into the tetrahedron of edge length n.

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