cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A378389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a tetrakis hexahedron.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 27 2024

Keywords

Comments

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

Examples

			2.498091544796508851659834154562180246155658808...

Links

Crossrefs

Cf. A378388 (surface area), A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1).
Cf. A156546 and A195698 (dihedral angles of a truncated octahedron), A195729.

Programs

  • Mathematica
    First[RealDigits[ArcCos[-4/5], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TetrakisHexahedron", "DihedralAngles"]], 10, 100]]

Formula

Equals arccos(-4/5).
Equals 2*A195729. - Amiram Eldar, Nov 27 2024

A334114 Decimal expansion of volume of a sphenomegacorona (J88) with each edge of unit length.

Original entry on oeis.org

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

Views

Author

Violeta Hernández Palacios, Apr 14 2020

Keywords

Comments

A sphenomegacorona is one of the 92 regular-faced non-isogonal convex polyhedra first enumerated by Norman W. Johnson. It's built out of 2 squares and 12 equilateral triangles.
This number is algebraic, of unknown degree.
It appears that the minimal polynomial is 521578814501447328359509917696*x^32 - 985204427391622731345740955648*x^30 - 16645447351681991898880656015360*x^28 + 79710816694053483249372512649216*x^26 - 152195045391070538203422101864448*x^24 + 156280253448056209478031589244928*x^22 - 96188116617075838858708654227456*x^20 + 30636368373570166303441645731840*x^18 + 5828527077458909552923002273792*x^16 - 8060049780765551057159394951168*x^14 + 1018074792115156107372011716608*x^12 + 35220131544370794950945931264*x^10 + 327511698517355918956755959808*x^8 - 116978732884218191486738706432*x^6 + 10231563774949176791703149568*x^4 - 366323949299263261553952192*x^2 + 3071435678740442112675625. - Joerg Arndt, Apr 16 2020

Examples

			1.94810822885947280327067639...

Links

Crossrefs

Volumes of other Johnson solids: A179552, A179587, A179590.

Programs

  • Mathematica
    k := Root[-23 - 56 x + 200 x^2 + 304 x^3 - 776 x^4 + 240 x^5 +
   2000 x^6 - 5584 x^7 - 3384 x^8 + 17248 x^9 + 2464 x^10 -
   24576 x^11 + 1568 x^12 + 17216 x^13 - 3712 x^14 - 4800 x^15 +
   1680 x^16, 2];
{{0, 1/2, Sqrt[1 - k^2]}, {k, 1/2, 0}, {0, Sqrt[(3/4 - k^2)/(1 - k^2)] + 1/2, (1/2 - k^2)/Sqrt[1 - k^2]}, {1/2, 0, -Sqrt[1/2 + k - k^2]}, {0, (Sqrt[3/4 - k^2] (2 k^2 - 1))/((k^2 - 1) Sqrt[1 - k^2]) + 1/2, (k^4 - 1/2)/(1 - k^2)^(3/2)}};
v = Union[%, {1, -1, 1}*# & /@ %, {-1, 1, 1}*# & /@ %, {-1, -1,
  1}*# & /@ %];
f := {{2, 3, 12, 11}, {2, 3, 10, 9}, {3, 12, 5}, {3, 10, 5}, {12, 5,
  7}, {10, 5, 7}, {7, 12, 8}, {7, 10, 1}, {12, 8, 11}, {10, 1,
  9}, {8, 1, 7}, {8, 1, 6}, {8, 11, 6}, {1, 9, 6}, {11, 6, 4}, {9,
  6, 4}, {4, 11, 2}, {4, 9, 2}};
RealDigits[N[Volume[Polyhedron[v, f]], 20000]][[1]]

A384266 G.f. A(x) = (3*sqrt(1 - 8*x) - (1 - 4*x)) / (2*(1 - 8*x - 2*x^2)).

Original entry on oeis.org

1, 4, 22, 136, 892, 6064, 42232, 299296, 2149360, 15596992, 114138592, 841108096, 6234779584, 46448349952, 347541337984, 2610319254016, 19671552622336, 148689857920000, 1126905157115392, 8561360256526336, 65185363066289152, 497307750242234368, 3800975843189291008, 29100188150365757440
Offset: 0

Views

Author

Paul D. Hanna, Jun 06 2025

Keywords

Comments

Compare formula (2.a) to C(x) = exp( x*C(x) + Integral C(x) dx ), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = 1 + 4*x + 22*x^2 + 136*x^3 + 892*x^4 + 6064*x^5 + 42232*x^6 + 299296*x^7 + 2149360*x^8 + 15596992*x^9 + ...
where log(A(x)) = x*A(x) + Integral A(x) + 2*A(x)^2 dx,
also, A'(x)/A(x) = 2 * A(x) * (1 + A(x)) / (1 - x*A(x)).
SPECIFIC VALUE.
Let z be a shared zero of both (1 - 8*x - 2*x^2) and (3*sqrt(1 - 8*x) - (1 - 4*x)), where z = (3*sqrt(2) - 4)/2 = 0.1213203435..., then A(z) = 2*(3 + 2*sqrt(2))/3 = sqrt(2)/(3*z) = 3.8856180831... (=2*A179587).

Links

Crossrefs

Cf. A179587.

Programs

  • Mathematica
    CoefficientList[Series[(3*Sqrt[1 - 8*x] - (1 - 4*x)) / (2*(1 - 8*x - 2*x^2)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jun 07 2025 *)
CoefficientList[Series[4/(1 + 3*Sqrt[1 - 8*x] - 4*x), {x,0,30}],x] (* Vaclav Kotesovec, Jun 07 2025 *)
  • PARI
    {a(n) = my(A = (3*sqrt(1 - 8*x +x*O(x^n)) - (1 - 4*x)) / (2*(1 - 8*x - 2*x^2)) );
polcoef(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = (3*sqrt(1 - 8*x) - (1 - 4*x)) / (2*(1 - 8*x - 2*x^2)).
(2.a) A(x) = exp( x*A(x) + Integral A(x) + 2*A(x)^2 dx ).
(2.b) A(x) = exp( Integral 2*A(x)*(1 + A(x))/(1 - x*A(x)) dx ).
(2.c) A'(x) = 2 * A(x)^2 * (1 + A(x)) / (1 - x*A(x)).
(3.a) A(x) = (3*sqrt(1 + 8*x^2*A(x)^2) - (1 - 8*x*A(x))) / 2.
(3.b) x/Series_Reversion(x*A(x)) = (3*sqrt(1 + 8*x^2) - (1 - 8*x)) / 2.
(4.a) [x^(2*n+1)] 1/A(x)^(2*n) = 0 for n >= 0.
(4.b) [x^(2*n)] 1/A(x)^(2*n) = [x^(2*n-1)] -1/A(x)^(2*n-1) for n >= 1.
From Vaclav Kotesovec, Jun 07 2025: (Start)
G.f.: 4/(1 + 3*sqrt(1 - 8*x) - 4*x).
Recurrence: n*a(n) = 4*(4*n-3)*a(n-1) - 2*(31*n-48)*a(n-2) - 8*(2*n-3)*a(n-3).
a(n) ~ 3 * 2^(3*n+3) / (sqrt(Pi)*n^(3/2)). (End)
Previous Showing 21-23 of 23 results.

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