A010503 -id:A010503 - cp's OEIS frontend

A199589 Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.

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

Offset: 1

Frank M Jackson, Nov 08 2011

If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where g = 1.1371580... and is the greatest root of the equation: 2 + 6d - 6d^3 = 0. The value of f is given in A199590.

			1.13715804260325761283766795192009876258136039422990658596288796494425...

Index entries for algebraic numbers, degree 3.

Cf. A010503, A016051, A199220, A199221, A199590.

N[Reduce[2+6d-6d^3==0, d], 100]
RealDigits[(2/Sqrt[3]) * Cos[Pi/18], 10, 120][[1]] (* Amiram Eldar, Nov 22 2024 *)

real(polroots(6*x^3-6*x-2)[3]) \\ Charles R Greathouse IV, Nov 10 2011

polrootsreal(6*x^3-6*x-2)[3] \\ Charles R Greathouse IV, Apr 14 2014

Equals sqrt(4/3)*cos(Pi/18). - Charles R Greathouse IV, Nov 10 2011

Equals Product_{k>=1} (1 - (-1)^k/A016051(k)). - Amiram Eldar, Nov 22 2024

A199590 Decimal expansion (unsigned) of the greatest root of 6x^3 + 18x^2 + 12x + 2 = 0.

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Nov 08 2011

If the side lengths of a quadrilateral form a harmonic progression in the ratio 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) where d is the common difference between the denominators of the harmonic progression, then the triangle inequality condition requires that d be in the range f < d < g, where f = -0.257772801... and is the greatest root of the equation: 2 + 12d + 18d^2 + 6d^3 = 0. The value of g is given in A199589.

			-0.257772801031440844729449397270635822708944125700977319782314639395808...

Index entries for algebraic numbers, degree 3

Cf. A010503, A199220, A199221, A199589.

Mathematica
```
N[Reduce[2+12d+18d^2+6d^3==0, d], 100]
```

real(polroots(6*x^3+18*x^2+12*x+2)[3]) \\ Charles R Greathouse IV, Nov 10 2011

polrootsreal(6*x^3-18*x^2+12*x-2)[1] \\ Charles R Greathouse IV, Oct 27 2023

sqrt(4/3)*sin(Pi*2/9) - 1. - Charles R Greathouse IV, Nov 10 2011

a(99) corrected by Sean A. Irvine, Jul 25 2021

A274541 Decimal expansion of exp(sqrt(2)/2).

Original entry on oeis.org

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

Offset: 1

Johannes W. Meijer, Jun 27 2016

Define P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), n >= 1 and P(0) = 1 with x(2) = (sqrt(2) + 1) and x(n) = 1 for all other n.
We find that C2 = lim_{n->infinity} P(n) = exp(sqrt(2)/2).
The structure of the n!*P(n) formulas leads to the multinomial coefficients A036039.

			c = 2.02811498164747245110812611274635117517432509254...

G. C. Greubel, Table of n, a(n) for n = 1..10000
The Dev Team and Simon Plouffe, The Inverse Symbolic Calculator (ISC).
Index entries for transcendental numbers

Cf. A274540, A274542, A014176, A010503, A036039.

SetDefaultRealField(RealField(100)); Exp[Sqrt[2]/2]; // G. C. Greubel, Aug 19 2018

Digits := 140: evalf(exp(sqrt(2)/2)); # End program 1.
P := proc(n) : if n=0 then 1 else P(n) := expand((1/n)*(add(x(n-k)*P(k), k=0..n-1))) fi; end: x := proc(n): if n=2 then (sqrt(2)+1) else 1 fi: end:
Digits := 140: evalf(P(250)); # End program 2.

First@ RealDigits@ N[Exp[Sqrt[2]/2], 83] (* Michael De Vlieger, Jun 27 2016 *)

my(x=exp(sqrt(2)/2)); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; print1(d, ", ")) \\ Felix Fröhlich, Jun 27 2016

c = exp(sqrt(2)/2).

c = lim_{n->infinity} P(n), with P(n) = (1/n)*Sum_{k=0..n-1} x(n-k)*P(k), for n >= 1, and P(0) = 1, with x(2) = (1 + sqrt(2)), the silver mean A014176, and x(n) = 1 for all other n.

More digits from Jon E. Schoenfield, Mar 15 2018

A376867 Reduced numerators of Newton's iteration for 1/sqrt(2), starting with 1/2.

Original entry on oeis.org

1, 5, 355, 94852805, 1709678476417571835487555, 9994796326591347130392203807311551183419838794447313956622219314498503205

Offset: 0

Steven Finch, Oct 07 2024

An explicit formula for a(n) is not known, although it arises from a recurrence and the corresponding denominators are simply 2^(3^n) = A023365(n+1).

Next term is too large to include.

			a(1) = 5 because b(1) = (1/2)*(3/2 - 1/4) = 5/8.
1/2, 5/8, 355/512, 94852805/134217728, ... = a(n)/A023365(n+1).

Alois P. Heinz, Table of n, a(n) for n = 0..7
X. Gourdon and P. Sebah, Pythagoras' Constant.

Cf. A010503, A023365, A376870.

b:= proc(n) b(n):= `if`(n=0, 1/2, b(n-1)*(3/2-b(n-1)^2)) end:
a:= n-> numer(b(n)):
seq(a(n), n=0..5);  # Alois P. Heinz, Oct 07 2024

a[0]=1/2; a[n_]:=a[n-1](3/2-a[n-1]^2); Numerator[Array[a,6,0]] (* Stefano Spezia, Oct 15 2024 *)

from itertools import count, islice
def A376867_gen(): # generator of terms
    p = 1
    for k in count(0):
        yield p
        p *= ((3<<((3**k<<1)-1))-p**2)
A376867_list = list(islice(A376867_gen(),6)) # Chai Wah Wu, Oct 11 2024

a(n) is the reduced numerator of b(n) = b(n-1)*(3/2 - b(n-1)^2); b(0) = 1/2.
Limit_{n -> oo} a(n)/A023365(n+1) = 1/sqrt(2) = A010503.
a(n+1) = a(n)*(3*2^(2*3^n-1)-a(n)^2). - Chai Wah Wu, Oct 11 2024

A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			32.4346643636148951726751573735281216767216730121...

Eric Weisstein's World of Mathematics, Truncated Cube.
Wikipedia, Truncated cube.

Cf. A377299 (volume), A294968 (circumradius), A010503 (midradius - 1), A377296 (Dehn invariant, negated).

Cf. A002193, A002194, A010469, A010524.

First[RealDigits[2*(6 + Sqrt[72] + Sqrt[3]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCube", "SurfaceArea"], 10, 100]]

Equals 2*(6 + 6*sqrt(2) + sqrt(3)) = 2*(6 + 2*A002193 + A002194) = 12 + 2*A010524 + A010469.

A212886 Decimal expansion of 2/(3sqrt(3)) = 2sqrt(3)/9.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 29 2012

Consider any cubic polynomial f(x) = a(x - r)(x - (r + s))(x -(r + 2s)), where a, r, and s are real numbers with s > 0 and nonzero a; i.e., any cubic polynomial with three distinct real roots, of which the middle root, r + s, is equidistant (with distance s) from the other two. Then the absolute value of f's local extrema is |a|*s^3*(2*sqrt(3)/9). They occur at x = r + s +- s*(sqrt(3)/3), with the local maximum, M, at r + s - s*sqrt(3)/3 when a is positive and at r + s + s*sqrt(3)/3 when a is negative (and the local minimum, m, vice versa). Of course m = -M < 0.
A quadratic number with denominator 9 and minimal polynomial 27x^2 - 4. - Charles R Greathouse IV, Apr 21 2016
This constant is also the maximum curvature of the exponential curve, occurring at the point M of coordinates [x_M = -log(2)/2 = (-1/10)*A016655; y_M = sqrt(2)/2 = A010503]. The corresponding minimum radius of curvature is (3*sqrt(3))/2 = A104956 (see the reference Eric Billault and the link MathStackExchange). - Bernard Schott, Feb 02 2020

			0.384900179459750509672765853667971637098401167513417917345734...

Eric Billault, Walter Damin, Robert Ferréol et al., MPSI - Classes Prépas, Khôlles de Maths, Ellipses, 2012, exercice 17.07 pages 386, 389-390.

MathStackExchange, Apex of an Exponential Function, 2015.
Index entries for algebraic numbers, degree 2

Cf. A002194, A104956, A020760.

Cf. A010503, A016655.

RealDigits[2/(3*Sqrt[3]), 10, 105] (* T. D. Noe, May 31 2012 *)

default(realprecision, 1000); 2*sqrt(3)/9

(2/9)*sqrt(3) = (2/9)*A002194.

A229117 Numbers k where d/k reaches a new record, with d the distance from the k-th triangular number to the nearest square.

Original entry on oeis.org

2, 3, 13, 20, 37, 78, 119, 218, 457, 696, 1273, 2666, 4059, 7422, 15541, 23660, 43261, 90582, 137903, 252146, 527953, 803760, 1469617, 3077138, 4684659, 8565558, 17934877, 27304196, 49923733, 104532126, 159140519, 290976842

Offset: 1

Ralf Stephan, Sep 14 2013

nonn

Positions of records of A229118(n)/n.

The maximum of d/k appears to converge to sqrt(2)/2 (A010503), i.e., k*(k+1)/2 is not more than k*sqrt(2)/2 distant from a square.

			G.f. = 2*x + 3*x^2 + 13*x^3 + 20*x^4 + 37*x^5 + 78*x^6 + 119*x^7 + 218*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A010503, A229118.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(2 +x+10*x^2-5*x^3+11*x^4-19*x^5+x^6-2*x^7+3*x^8)/(1-x-6*x^3+6*x^4+x^6- x^7))); // G. C. Greubel, Aug 09 2018

Drop[CoefficientList[Series[x*(2 + x + 10*x^2 - 5*x^3 + 11*x^4 - 19*x^5 + x^6 - 2*x^7 + 3*x^8)/(1 - x - 6*x^3 + 6*x^4 + x^6 - x^7), {x, 0, 50}], x], 1] (* G. C. Greubel, Aug 09 2018 *)

m=0;for(n=1, 10^9, t=n*(n+1)/2;s=sqrtint(t);d=min(t-s^2,(s+1)^2-t);r=d/n;if(r>m,m=r;print1(n, ",")))

{a(n) = if( n<1, 0, polcoeff( (1 + x + x^2 + 4*x^3 + x^4 + 11*x^5 - 18*x^6 - 2*x^8 + 3*x^9) / (1 - x - 6*x^3 + 6*x^4 + x^6 - x^7) + x * O(x^n), n))}; /* Michael Somos, Dec 25 2016 */

G.f.: x * (2 + x + 10*x^2 - 5*x^3 + 11*x^4 - 19*x^5 + x^6 - 2*x^7 + 3*x^8) / (1 - x - 6*x^3 + 6*x^4 + x^6 - x^7). - Michael Somos, Dec 25 2016

a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7) if n>9. - Michael Somos, Dec 25 2016

A237129 Let d = d(1)d(2)... d(q) denote the decimal expansion of an angle d expressed in degrees. The sequence a(n) lists the angles such that sin(d) = cos(d(1)d(2)... *d(q)).

Original entry on oeis.org

90, 418, 450, 666, 726, 778, 786, 810, 1146, 1170, 1386, 1395, 1530, 1775, 1890, 2218, 2250, 2394, 2474, 2482, 2610, 2842, 2898, 2970, 3186, 3195, 3312, 3330, 3366, 3375, 3690, 3711, 3735, 3915, 3933, 3978, 4050, 4146, 4194, 4274, 4282, 4338, 4410, 4698, 4770

Offset: 1

Michel Lagneau, Feb 04 2014

			666 is in the sequence because sin(666°) = cos(6*6*6°) = -.8090169943749... = -phi/2 where phi is the golden ratio (1+sqrt(5))/2. (A019863)
418 is in the sequence because sin(418°) = cos(4*1*8°)= .84804809615... (A019867)
3915 is in the sequence because sin(3915°) = cos(3*9*1*5°)= -.70710678118654752440 = -1/sqrt(2). (A010503)

Michel Lagneau, Table of n, a(n) for n = 1..2000

with(numtheory):err:=1/10^10:Digits:=20:for n from 1 to 5000 do:x:=convert(n,base,10):n1:=nops(x):p:=product('x[i]', 'i'=1..n1):s1:=evalf(sin(n*Pi/180)):s2:=evalf(cos(p*Pi/180)):if abs(s1-s2)

A263353 Decimal expansion of the generalized hypergeometric function 3F2(1/2,1/2,1/2; 3/2,3/2; x) at x=1/2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 16 2015

			1.032631955744071472677093...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

G. C. Greubel, Table of n, a(n) for n = 1..10000
R. J. Mathar, Yet another table of integrals, arXiv:1207.5845 [math.CA], 2012-2016. Eq. (9.81).
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 65.

Cf. A002162, A003881, A006752, A010503.

SetDefaultRealField(RealField(100)); R:=RealField(); (Pi(R)*Log(2)/4 + Catalan(R))/Sqrt(2); // G. C. Greubel, Aug 25 2018

evalf(hypergeom([1/2,1/2,1/2],[3/2,3/2],1/2) );

RealDigits[(Pi*Log[2]/4 + Catalan)/Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

default(realprecision, 100); (Pi*log(2)/4 + Catalan)/sqrt(2) \\ G. C. Greubel, Aug 25 2018

Equals (Pi*log(2)/4+Catalan)/sqrt(2) = (A003881 * A002162 + A006752) * A010503.

Equals Sum_{k>=0} binomial(2*k,k)/(2^(3*k)*(2*k + 1)^2) (see Finch). - Stefano Spezia, Nov 12 2024

A270394 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r(k) = 1/Fibonacci(k+1).

Original entry on oeis.org

2, 3, 9, 59, 9437, 62059971, 2813586350787717, 8534689167911295735140758101600, 54171527001975050997893888972139886506909953999125751170768531

Offset: 1

Clark Kimberling, Mar 22 2016

Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1) + r(2)/n(2) + r(3)/n(3) + ..., the r-Egyptian fraction for x.

See A269993 for a guide to related sequences.

			sqrt(1/2) = 1/2 + 1/(2*3) + 1/(3*9) + 1/(5*59) + ...

Clark Kimberling, Table of n, a(n) for n = 1..12
Eric Weisstein's World of Mathematics, Egyptian Fraction
Index entries for sequences related to Egyptian fractions

Cf. A269993, A000045, A010503.

r[k_] := 1/Fibonacci[k+1]; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]

r(k) = 1/fibonacci(k+1);
f(k,x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x););
a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016

cp's OEIS Frontend

A199589 Decimal expansion of the greatest root of 6x^3 - 6x - 2 = 0.

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A199590 Decimal expansion (unsigned) of the greatest root of 6x^3 + 18x^2 + 12x + 2 = 0.

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A274541 Decimal expansion of exp(sqrt(2)/2).

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A376867 Reduced numerators of Newton's iteration for 1/sqrt(2), starting with 1/2.

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A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

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A212886 Decimal expansion of 2/(3*sqrt(3)) = 2*sqrt(3)/9.

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A229117 Numbers k where d/k reaches a new record, with d the distance from the k-th triangular number to the nearest square.

A212886 Decimal expansion of 2/(3sqrt(3)) = 2sqrt(3)/9.

A237129 Let d = d(1)d(2)... d(q) denote the decimal expansion of an angle d expressed in degrees. The sequence a(n) lists the angles such that sin(d) = cos(d(1)d(2)... *d(q)).