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

Author

Felix Tubiana, Feb 05 2004

Keywords

• easy
• nonn
• cons

Comments

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)

Examples

			3.79128784747792...
		

References

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

Links

• 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

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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

PARI

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

Formula

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

Extensions

Additional comments from Rick L. Shepherd, Jul 02 2004