A356034 -id:A356034 - cp's OEIS frontend

A090458 Decimal expansion of (3 + sqrt(21))/2.

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

Offset: 1

Felix Tubiana, Feb 05 2004

Decimal expansion of the solution to n/x = x-n for n-3. n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...
n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...
x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1,...]. (See the Mathematica program below.) - Clark Kimberling, May 05 2011
x appears to be the limit for n to infinity of the ratio of the number of even numbers that take n steps to reach 1 to the number of odd numbers that take n steps to reach 1 in the Collatz iteration. As A005186(n-1) is the number of even numbers that take n steps to reach 1, this means x = lim A005186(n-1)/A176866(n). - Markus Sigg, Oct 20 2020
From Wolfdieter Lang, Sep 02 2022: (Start)
This integer in the quadratic number field Q(sqrt(21)) equals the (real) cube root of 27 + 6*sqrt(21) = 54.4954541... . See Euler, Elements of Algebra, Article 748 or Algebra (in German) p. 306, Kapitel 12, 187.
Subtracting 3 from the present number gives the (real) cube root of
  -27 +  6*sqrt(21) = 0.4954541... . (End)

			3.79128784747792...

Leonhard Euler, Vollständige Anleitung zur Algebra, (1770), Reclam, Leipzig, 1883, p.306, Kapitel 12, 187.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Leonhard Euler, Elements of Algebra, p. 244, Article 748.
Markus Sigg, Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration, 2020.
Index entries for algebraic numbers, degree 2

Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).
Equals 3*A176014 (constant).
Cf. A356034.

FromContinuedFraction[{3,1,{3,1}}]
ContinuedFraction[%,20]
RealDigits[N[%%,120]] (*A090458*)
N[%%%,40]
RealDigits[(3 + Sqrt[21])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

solve(x=3,4,x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011

(3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011

Equals (27 + 6*sqrt(21))^(1/3). - Wolfdieter Lang, Sep 01 2022

Additional comments from Rick L. Shepherd, Jul 02 2004

A357100 Decimal expansion of the real root of x^3 + x^2 - 3.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Sep 13 2022

This equals r0 - 1/3 where r0 is the real root of y^3 - (1/3)*y - 79/27.
The other two roots are (w1*(79/2 + (9/2)*sqrt(77))^(1/3) + w2*(79/2 - (9/2)*sqrt(77))^(1/3) - 1)/3 = -1.08727970... + 1.1713121...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic function these roots are (-(1 + cosh((1/3)*arccosh(79/2))) + sqrt(3)*sinh((1/3)*arccosh(79/2))*i)/3, and its complex conjugate.

			r = 1.17455941029298007420231898869565392567594872533708249833673392030236...

Cf. A356034.

Digits := 120: a := ((79 + 9*sqrt(77))/2)^(1/3): (a + 1/a - 1)/3: evalf(%)*10^96: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 15 2022

First[RealDigits[x/.N[First[Solve[x^3+x^2-3==0, x]], 76]]] (* Stefano Spezia, Sep 15 2022 *)

(2*cosh((1/3)*acosh(79/2)) - 1)/3 \\ Michel Marcus, Sep 15 2022

r = ((316 + 36*sqrt(77))^(1/3) + 4/(316 + 36*sqrt(77))^(1/3) - 2)/6.
r = ((79/2 + (9/2)*sqrt(77))^(1/3) + (79/2 - (9/2)*sqrt(77))^(1/3) - 1)/3.
r = (2*cosh((1/3)*arccosh(79/2)) - 1)/3.

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A090458 Decimal expansion of (3 + sqrt(21))/2.

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A357100 Decimal expansion of the real root of x^3 + x^2 - 3.

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