A187799 -id:A187799 - cp's OEIS frontend

A239798 Decimal expansion of the midsphere radius in a regular dodecahedron with unit edges.

1, 3, 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

Offset: 1

Stanislav Sykora, Mar 27 2014

In a regular polyhedron, the midsphere is tangent to all edges.

Apart from leading digits the same as A019863 and A019827. - R. J. Mathar, Mar 30 2014

			1.30901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A000045, A001622, A001654, A019827, A019863, A066983, A094884, A104457, A187799.

Midsphere radii in Platonic solids: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A019863 (icosahedron).

Digits:=100: evalf((3+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[GoldenRatio^2/2,10,105][[1]] (* Vaclav Kotesovec, Mar 27 2014 *)

PARI
```
(3+sqrt(5))/4
```

Equals phi^2/2, phi being the golden ratio (A001622).
Equals (3+sqrt(5))/4.
Equals lim_{n->oo} A000045(n)/A066983(n). - Dimitri Papadopoulos, Nov 23 2023
Equals Product_{k>=2} (1 + (-1)^k/A001654(k)). - Amiram Eldar, Dec 02 2024
Equals A094884^2 = A104457/2 = 10/A187799. - Hugo Pfoertner, Dec 02 2024

A244847 Decimal expansion of rho_c = (5-sqrt(5))/10, the asymptotic critical density for the hard hexagon model.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 12 2014

The vertical distance between the accumulation point and the outermost point of a golden spiral inscribed inside a golden rectangle with dimensions phi and 1 along the x and y axes, respectively (the horizontal distance is A176015). - Amiram Eldar, May 18 2021

The limiting frequency of the digit 1 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			0.2763932022500210303590826331268723764559381640388474275729102754589479...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.12.1 Phase transitions in Lattice Gas Models, p. 347.

Rodney J. Baxter Hard hexagons: exact solution, Journal of Physics A: Mathematical and General, Vol. 13, No. 3 (1980), pp. L61-L70, alternative link.
P. S. Bruckman and I. J. Good, A Generalization of a Series of De Morgan with Applications of Fibonacci Type, The Fibonacci Quarterly, Vol. 14, No. 3 (1976), pp. 193-196.
Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Eric Weisstein's MathWorld, Hard Hexagon Entropy Constant.
Wikipedia, Hard Hexagon Model.
Index entries for algebraic numbers, degree 2.

Cf. A242671, A244593.
Cf. A000032, A000045.
Essentially the same sequence of digits as A229760 and A187799.

RealDigits[(5 - Sqrt[5])/10, 10, 102] // First

Equals 1/(sqrt(5)*phi), where phi = (1+sqrt(5))/2 = A001622. - Vaclav Kotesovec, Nov 13 2014
Equals lim_{n -> infinity} A000045(n)/A000032(n+1). - Bruno Berselli, Jan 22 2018
Equals Sum_{n>=1} A000045(3^(n-1))/A000032(3^n) = Sum_{n>=1} A045529(n-1)/A006267(n). - Amiram Eldar, Dec 20 2018
Equals 1 - A242671. - Amiram Eldar, Mar 18 2025

A229760 Decimal expansion of 25 - 10*sqrt(5).

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Sep 28 2013

Apart from the first digit the same as A187799.

			2.639320225002103035908263312687237645593816403884742757291027545894790...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A187426, A187798, A094874.

RealDigits[25 - Sqrt@ 500, 10, 111][[1]] (* Robert G. Wilson v, Oct 01 2013 *)

25-sqrt(500) \\ Charles R Greathouse IV, Oct 01 2013

A229759 Decimal expansion of (25-10*sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Joost Gielen, Sep 28 2013

Essentially the same as A225667 and A132338. - R. J. Mathar, Sep 30 2013

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A187426, A187798, A094874, A187799.

(25-10*sqrt(5))/2 = 25/2 - 5*sqrt(5) = 1.319660... .

A255353 Denominators in an expansion of 3 - sqrt(5) as a sum of fractions +-1/d.

Original entry on oeis.org

2, 3, 6, 15, 24, 40, 104, 168, 273, 714, 1155, 1870, 4895, 7920, 12816, 33552, 54288, 87841, 229970, 372099, 602070, 1576239, 2550408, 4126648, 10803704, 17480760, 28284465, 74049690, 119814915, 193864606, 507544127, 821223648, 1328767776

Offset: 1

Mohammad K. Azarian, Feb 21 2015

The minus sign in front of a fraction is considered the sign of the numerator and hence the sign of the fraction does not appear in this sequence. We note that numerators are in A131561.

			1/(1*2) + 1/(1*3) - 1/(2*3) + 1/(3*5) + 1/(3*8) - 1/(5*8) + 1/(8*13) + 1/(8*21) - 1/(13*21) + 1/(21*34) + 1/(21*55) - 1/(34*55) + ... + = 3 - sqrt(5).

Colin Barker, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, The Value of a Series of Reciprocal Fibonacci Numbers, Problem B-1133, Fibonacci Quarterly, Vol. 51, No. 3, August 2013, p. 275; Solution published in Vol. 52, No. 3, August 2014, pp. 277-278.
Index entries for linear recurrences with constant coefficients, signature (0,0,8,0,0,-8,0,0,1).

Cf. A131561, A187799.

Table[SeriesCoefficient[x (2 + 3 x + 6 x^2 - x^3 - 8 x^5 + x^8)/((1 - x) (1 + x + x^2) (1 - 7 x^3 + x^6)), {x, 0, n}], {n, 33}] (* Michael De Vlieger, Dec 17 2015 *)

Vec(x*(2+3*x+6*x^2-x^3-8*x^5+x^8)/((1-x)*(1+x+x^2)*(1-7*x^3+x^6)) + O(x^40)) \\ Colin Barker, Dec 17 2015

3 - sqrt(5) = Sum_{n>=1} 1/(F(2*n)*F(2*n+1)) + 1/(F(2*n)*F(2*n+2)) - 1/(F(2*n+1)*F(2*n+2)), where F = A000045 (Fibonacci numbers).
From Colin Barker, Dec 17 2015: (Start)
a(n) = 8*a(n-3) - 8*a(n-6) + a(n-9) for n>9.
G.f.: x*(2+3*x+6*x^2-x^3-8*x^5+x^8) / ((1-x)*(1+x+x^2)*(1-7*x^3+x^6)).
(End)

A262353 a(n) = ceiling((3-sqrt(5))10^(2n+1)).

Original entry on oeis.org

8, 764, 76394, 7639321, 763932023, 76393202251, 7639320225003, 763932022500211, 76393202250021031, 7639320225002103036, 763932022500210303591, 76393202250021030359083, 7639320225002103035908264, 763932022500210303590826332, 76393202250021030359082633127

Offset: 0

Martin Renner, Mar 24 2016

a(n) is a special family of 2nd-order base-10 grafting integers, because every integer generated by ceiling((3-sqrt(5))*10^(2*n+1)) is a grafting integer.
A grafting number is a number whose digits, represented in base b, appear before or directly after the decimal point of its r-th root. Numbers of the simplest type deal with square roots in the decimal system.
The constant x = 3-sqrt(5) is a solution of the general grafting equation (x*b^a)^(1/r) = x + c with corresponding values r = 2, b = 10, a = 1, c = 2 (where r >= 2 is the grafting root, b >= 2 is the base in which the numbers are represented, a >= 0 is the number of places the decimal point is shifted, and c >= 0 is the constant added to the front of the result).

			sqrt(8) = 2.828427...,
sqrt(764) = 27.6405...,
sqrt(76394) = 276.39464...

Matt Parker, Things to make and do in the Fourth Dimension, New York (Ferrar, Strauss and Giroux), 2014, p. 62-63.

Subsequence of A232087.

Cf. A187799.

[Ceiling((3-Sqrt(5))*10^(2*n+1)):n in [0..14]]; // Marius A. Burtea, Aug 08 2019

Digits:=2000: a:=n->ceil((3-sqrt(5))*10^(2*n+1)); seq(a(n),n=0..14);

Table[Ceiling[(3 - Sqrt@ 5) 10^(2 n + 1)], {n, 14}] (* Michael De Vlieger, Mar 24 2016 *)

a(n) = ceil((3-sqrt(5))*10^(2*n+1)); \\ Altug Alkan, Mar 24 2016

a(n) = 30*100^n - sqrtint(10^(4*n+2)*5) \\ Charles R Greathouse IV, Jan 20 2017

a(n) = ceiling((3-sqrt(5))*10^(2*n+1)).

a(0) = 8 prepended by Robert Tanniru, Aug 06 2019

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A239798 Decimal expansion of the midsphere radius in a regular dodecahedron with unit edges.

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A244847 Decimal expansion of rho_c = (5-sqrt(5))/10, the asymptotic critical density for the hard hexagon model.

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A229760 Decimal expansion of 25 - 10*sqrt(5).

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A229759 Decimal expansion of (25-10*sqrt(5))/2.

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A255353 Denominators in an expansion of 3 - sqrt(5) as a sum of fractions +-1/d.

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A262353 a(n) = ceiling((3-sqrt(5))*10^(2*n+1)).

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A262353 a(n) = ceiling((3-sqrt(5))10^(2n+1)).