A134946 -id:A134946 - cp's OEIS frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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

Offset: 0

N. J. A. Sloane

Midsphere radius of regular icosahedron with unit edges.
Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014
Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014
This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - Wolfdieter Lang, Jan 08 2018
Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Feb 24 2021

			0.80901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014).
Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008).
Irit Dinur and David Steurer, Analytical approach to parallel repetition, arXiv:1305.1979 [cs.CC], 2013-2014.
Michael Penn, a golden value of cosine., YouTube video, 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

convert(sin(3*Pi/10),radical); # W. Edwin Clark, May 24 2023
Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[54 Degree],10,120][[1]] (* Harvey P. Dale, Apr 21 2018 *)

(1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - R. J. Mathar, Jun 18 2006
Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008
Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014
phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014
Equals 1/A134972. - R. J. Mathar, Jan 17 2021
Equals 2*A019836*A019872. - R. J. Mathar, Jan 17 2021
Equals (A094214 + 1)/2 or 1/(2*A094214). - Burak Muslu, Feb 24 2021
Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - Peter Bala, Mar 04 2022
Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - Amiram Eldar, Nov 28 2024
Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - Hugo Pfoertner, Nov 28 2024
Equals A134945/4. Root of 4*x^2-2*x-1=0. - R. J. Mathar, Aug 29 2025

A034896 Number of solutions to a^2 + b^2 + 3c^2 + 3d^2 = n.

Original entry on oeis.org

1, 4, 4, 4, 20, 24, 4, 32, 52, 4, 24, 48, 20, 56, 32, 24, 116, 72, 4, 80, 120, 32, 48, 96, 52, 124, 56, 4, 160, 120, 24, 128, 244, 48, 72, 192, 20, 152, 80, 56, 312, 168, 32, 176, 240, 24, 96, 192, 116, 228, 124, 72, 280, 216, 4, 288, 416, 80, 120, 240, 120, 248, 128, 32, 500

Offset: 0

N. J. A. Sloane

nonn

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Number 16 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

			G.f. = 1 + 4*x + 4*x^2 + 4*x^3 + 20*x^4 + 24*x^5 + 4*x^6 + 32*x^7 + ... - _Michael Somos_, Nov 10 2018

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223, Entry 3(iv).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Gilleland, Some Self-Similar Integer Sequences
J. Liouville, Sur la forme x^2 + y^2 + 3(z^2 + t^2), Journal de mathématiques pures et appliquées 2e série, tome 5 (1860), p. 147-152.
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A272364, A320147, A320148.

Cf. A033716, A113262, A134946.

A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* Michael Somos, Nov 10 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^10 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^4, n))}; /* Michael Somos, Nov 10 2018 */

Expansion of theta_3(q)^2*theta_3(q^3)^2.
G.f.: s(2)^10*s(6)^10/(s(1)*s(3)*s(4)*s(12))^4, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
Fine gives an explicit formula for a(n) in terms of the divisors of n.
From Michael Somos, Nov 10 2018: (Start)
Expansion of (a(q) + 2*a(q^4))^2 / 9 = (a(q)^2 - 2*a(q^2)^2 + 4*a(q^4)^2) / 3 in powers of q where a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + 4 Sum_{k>0} k x^k / (1 - (-x)^k) Kronecker(9, k).
a(n) = 1 + 4 * A113262(n) = (-1)^n * A134946(n). Convolution square of A033716.
a(n) = 4 * (s(n) - 2*s(n/2) - 3*s(n/3) + 4*s(n/4) + 6*s(n/6) - 12*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0. (End)

A377805 Decimal expansion of the volume of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 09 2024

			37.616649962733362975777673671302714340355289873...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.
Index entries for algebraic numbers, degree 12.

Cf. A377804 (surface area), A377806 (circumradius), A377807 (midradius).
Cf. A102769 (analogous for a regular dodecahedron).
Cf. A001622, A090550, A104457, A134946, A377849.

First[RealDigits[((3*GoldenRatio + 1)*#*(# + 1) - GoldenRatio/6 - 2)/Sqrt[3*#^2 - GoldenRatio^2], 10, 100]] & [Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "Volume"], 10, 100]]

Equals ((3*phi + 1)*xi*(xi + 1) - phi/6 - 2)/sqrt(3*xi^2 - phi^2) = (A090550*xi*(xi + 1) - A134946 - 2)/sqrt(3*xi^2 - A104457), where phi = A001622 and xi = A377849.

Equals the largest real root of 2176782336*x^12 - 3195335070720*x^10 + 162223191936000*x^8 + 1030526618040000*x^6 + 6152923794150000*x^4 - 182124351550575000*x^2 + 187445810737515625.

A386691 Decimal expansion of the volume of a parabidiminished rhombicosidodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 30 2025

The parabidiminished rhombicosidodecahedron is Johnson solid J_80.

Also the volume of a metabidiminished rhombicosidodecahedron and a gyrate bidiminished rhombicosidodecahedron (Johnson solids J_81 and J_82, respectively) with unit edges.

			36.967233145831580803409780572760635295338486330...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyrate bidiminished rhombicosidodecahedron.
Wikipedia, Metabidiminished rhombicosidodecahedron.
Wikipedia, Parabidiminished rhombicosidodecahedron.

Cf. A386692 (surface area).

Cf. A001622, A002163, A134946, A179590, A185093, A386689, A386693.

First[RealDigits[5/3*(11 + 5*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J80", "Volume"], 10, 100]]

Equals (5/3)*(11 + 5*sqrt(5)) = (5/3)*(11 + 5*A002163).
Equals A185093 - 2*A179590.
Equals (50/3)*A001622 + 10 = A134946*100 + 10.
Equals the largest root of 9*x^2 - 330*x - 100.

A384682 Decimal expansion of (5/6)phi = 5(1 + sqrt(5))/12, where phi is the golden ratio.

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, Jun 06 2025

			1.34836165729157904017...

Cf. A001622, A021016, A134944, A134946, A384238.

RealDigits[GoldenRatio * 5/6, 10, 100, 0][[1]]

Minimal polynomial: 36*x^2 - 30*x - 25.
Equals Integral_{x=0..1} sqrt(x + sqrt(x + sqrt(x + ...))) dx.
Equals Integral_{x=0..1} (1 + sqrt(1 + 4*x))/2 dx.
Equals 10*A134944/3 = 5*A134946. - Hugo Pfoertner, Jun 07 2025

A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

Original entry on oeis.org

1, 1, 2, 2, 1, 6, 2, 2, 6, 5, 4, 4, 1, 10, 8, 4, 3, 2, 8, 14, 2, 12, 10, 4, 11, 5, 14, 10, 4, 28, 8, 8, 8, 1, 20, 2, 7, 4, 8, 14, 6, 6, 18, 8, 24, 6, 2, 4, 22, 31, 12, 14, 9, 10, 2, 12, 16, 12, 20, 20, 5, 8, 8, 20, 13, 20, 22, 2, 10, 52, 28, 2, 15, 19, 36, 4

Offset: 1

Vanessa Gomez, Eric Jovinelly, Jacob A. McCann, Bryce Orloski, Catherine Rea, Shannon Talbott, Jul 17 2017

We calculated the continued fraction expansion of phi/n and observed that the expansion is periodic after the first nonzero term. We tracked the periodicity of the expansions and present them here. The authors acknowledge the National Science Foundation (DMS-1560019) and Muhlenberg College for supporting the REU (Research Experiences for Undergraduates) on which this sequence is based.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001622 (phi), A019863 (phi/2), A134943 (phi/3), A134944 (phi/4), A134946 (phi/6).

a[n_] := ContinuedFraction[GoldenRatio/n] // Last // Length; Array[a, 80] (* Jean-François Alcover, Jul 28 2017 *)

cp's OEIS Frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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A034896 Number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

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A377805 Decimal expansion of the volume of a snub dodecahedron with unit edge length.

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A386691 Decimal expansion of the volume of a parabidiminished rhombicosidodecahedron with unit edges.

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A384682 Decimal expansion of (5/6)*phi = 5*(1 + sqrt(5))/12, where phi is the golden ratio.

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A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

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A034896 Number of solutions to a^2 + b^2 + 3c^2 + 3d^2 = n.

A384682 Decimal expansion of (5/6)phi = 5(1 + sqrt(5))/12, where phi is the golden ratio.