A199589 -id:A199589 - cp's OEIS frontend

A016051 Numbers of the form 9k+3 or 9k+6.

3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 111, 114, 120, 123, 129, 132, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 192, 195, 201, 204, 210, 213, 219, 222, 228, 231, 237, 240

Offset: 1

Robert G. Wilson v

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A199589.

Subsequence of A145204. - Reinhard Zumkeller, Oct 04 2008

Select[Range[240], MatchQ[Mod[#, 9], 3|6]&] (* Jean-François Alcover, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{3,6,12},60] (* or *) #+{3,6}&/@(9*Range[0,30])//Flatten (* Harvey P. Dale, Oct 04 2021 *)

a(n) = 3*A001651(n).
a(n+1) = a(n) + its digital root in decimal base.
From R. J. Mathar, Dec 16 2009: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) = 9*n/2 - 9/4 - 3*(-1)^n/4.
G.f: 3*x*(1+x+x^2)/((1+x)*(x-1)^2). (End)
a(n) = 9*(n-1) - a(n-1) (with a(1)=3). - Vincenzo Librandi, Nov 19 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(9*sqrt(3)). - Amiram Eldar, Sep 26 2022
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = (2/sqrt(3)) * cos(Pi/18) (A199589).
Product_{n>=1} (1 + (-1)^n/a(n)) = (2/sqrt(3)) * sin(2*Pi/9). (End)

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

A199861 The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Nov 11 2011

Brahmagupta expression sqrt((-1+1/(1+d)+1/(1+2d)+1/(1+3d)) * (1-1/(1+d)+1/(1+2d)+1/(1+3d)) * (1+1/(1+d)-1/(1+2d)+1/(1+3d)) * (1+1/(1+d)+1/(1+2d)-1/(1+3d)))/4 for d in the interval [-1/3, inf] where 1/(1+d), 1/(1+2d) and 1/(1+3d) are always positive.

The area of a convex quadrilateral with fixed sides is maximal when it is organized as a convex cyclic quadrilateral. Furthermore in order that a quadrilateral can have sides in a harmonic progression 1 : 1/(1+d) : 1/(1+2d) : 1/(1+3d) its denominator's common difference d is limited to the range f < d < g where f is the constant A199590 and g is the constant A199589. Consequently when d=-0.2271064482... it maximizes Brahmagupta's expression for the area of a convex cyclic quadrilateral whose sides form a harmonic progression.

			-0.22710644829438120301114335253234461837754...

Wikipedia, Brahmagupta's formula.
Index entries for algebraic numbers, degree 12

Cf. A010503, A193180, A193211, A199589, A199590.

RealDigits[d/.NMaximize[{Sqrt[(-1+1/(1+d)+1/(1+2d)+1/(1+3d))(1-1/(1+d)+1/(1+2d)+1/(1+3d))(1+1/(1+d)-1/(1+2d)+1/(1+3d))(1+1/(1+d)+1/(1+2d)-1/(1+3d))]/4, -1/4120, PrecisionGoal->100, WorkingPrecision->240][[2]]][[1]]

real(polroots(1323*d^12 + 9711*d^11 + 32535*d^10 + 67005*d^9 + 94338*d^8 + 94761*d^7 + 68955*d^6 + 36367*d^5 + 13740*d^4 + 3619*d^3 + 630*d^2 + 65*d + 3)[4]) \\ Charles R Greathouse IV, Nov 11 2011

polrootsreal(1323*x^12 - 9711*x^11 + 32535*x^10 - 67005*x^9 + 94338*x^8 - 94761*x^7 + 68955*x^6 - 36367*x^5 + 13740*x^4 - 3619*x^3 + 630*x^2 - 65*x + 3)[1] \\ Charles R Greathouse IV, Oct 27 2023

d is the largest real root of the equation 1323d^12 + 9711d^11 + 32535d^10 + 67005d^9 + 94338d^8 + 94761d^7 + 68955d^6 + 36367d^5 + 13740d^4 + 3619d^3 + 630d^2 + 65d + 3 = 0.

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A016051 Numbers of the form 9k+3 or 9k+6.

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

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A199861 The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

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cp's OEIS Frontend

A016051 Numbers of the form 9*k+3 or 9*k+6.

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A199861 The decimal expansion (unsigned) of the value of d that maximizes the Brahmagupta expression given below.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A016051 Numbers of the form 9k+3 or 9k+6.