

A090458


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


14



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
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OFFSET

1,1


COMMENTS

Decimal expansion of the solution to n/x = xn for n3. n/x = xn with n=1 gives the Golden Ratio = 1.6180339887...
n/x = xn ==> 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(n1) is the number of even numbers that take n steps to reach 1, this means x = lim A005186(n1)/A176866(n).  Markus Sigg, Oct 20 2020


LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Markus Sigg, Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration, 2020.


EXAMPLE

3.79128784747792...


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 *)


PROG

(PARI) solve(x=3, 4, x^23*x3) \\ Charles R Greathouse IV, Oct 04 2011
(PARI) (3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011


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).
Sequence in context: A217359 A244338 A336045 * A131712 A072845 A197481
Adjacent sequences: A090455 A090456 A090457 * A090459 A090460 A090461


KEYWORD

easy,nonn,cons


AUTHOR

Felix Tubiana (fat2(AT)columbia.edu), Feb 05 2004


EXTENSIONS

Additional comments from Rick L. Shepherd, Jul 02 2004


STATUS

approved



