A090458 - OEIS

%I #72 Aug 21 2023 10:19:36

%S 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,

%T 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,

%U 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

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

%C 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...

%C 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...

%C 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

%C 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

%C From _Wolfdieter Lang_, Sep 02 2022: (Start)

%C 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.

%C Subtracting 3 from the present number gives the (real) cube root of

%C   -27 +  6*sqrt(21) = 0.4954541... . (End)

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

%H Chai Wah Wu, <a href="/A090458/b090458.txt">Table of n, a(n) for n = 1..10000</a>

%H Leonhard Euler, <a href="https://archive.org/details/ElementsOfAlgebraLeonhardEuler2015/page/244/mode/1up?view=theater">Elements of Algebra</a>, p. 244, Article 748.

%H Markus Sigg, <a href="/A005186/a005186_4.pdf">Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration</a>, 2020.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>

%F Equals (27 + 6*sqrt(21))^(1/3). - _Wolfdieter Lang_, Sep 01 2022

%e 3.79128784747792...

%t FromContinuedFraction[{3,1,{3,1}}]

%t ContinuedFraction[%,20]

%t RealDigits[N[%%,120]] (*A090458*)

%t N[%%%,40]

%t RealDigits[(3 + Sqrt[21])/2, 10, 50][[1]] (* _G. C. Greubel_, Jul 03 2017 *)

%o (PARI) solve(x=3,4,x^2-3*x-3) \\ _Charles R Greathouse IV_, Oct 04 2011

%o (PARI) (3+sqrt(21))/2 \\ _Charles R Greathouse IV_, Oct 04 2011

%Y 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).

%Y Equals 3*A176014 (constant).

%Y Cf. A356034.

%K easy,nonn,cons

%O 1,1

%A _Felix Tubiana_, Feb 05 2004

%E Additional comments from _Rick L. Shepherd_, Jul 02 2004