A002947 -id:A002947

Search: a002947 -id:a002947

Displaying 1-2 of 2 results found.

page 1

A005480

Decimal expansion of cube root of 4.
(Formerly M3771)

+10
20

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Let h = 4^(1/3). Then (h+1,0) is the x-intercept of the shortest segment from the x-axis through (1,2) to the y-axis; see A197008. - Clark Kimberling, Oct 10 2011

Let h = 4^(1/3). The relative maximum of xy(x+y)=1 is (-1/sqrt(h), h). - Clark Kimberling, Oct 05 2020

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data, Scripta Math. 18, (1952), p. 173-176.

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000

Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data, Scripta Math. 18, (1952). 173-176. [Annotated scanned copies of pages 175 and 176 only]

Index entries for algebraic numbers, degree 3

FORMULA

Equals Product_{k>=0} (1 + (-1)^k/(3*k + 1)). - Amiram Eldar, Jul 25 2020

Equals A002580^2. - Michel Marcus, Jan 08 2022

Equals hypergeom([1/3, 1/6], [2/3], 1). - Peter Bala, Mar 02 2022

EXAMPLE

1.587401051968199474751705639272308260391493327899853...

MATHEMATICA

RealDigits[N[4^(1/3), 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

PROG

(PARI) default(realprecision, 20080); x=4^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005480.txt", n, " ", d)); \\ Harry J. Smith, May 07 2009, with a correction made May 19 2009

CROSSREFS

Cf. A002947 (continued fraction). - Harry J. Smith, May 07 2009

Cf. A002580 (cube root of 2).

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane; entry revised Apr 23 2006

STATUS

approved

A002945

Continued fraction for cube root of 2.
(Formerly M2220)

+10
16

1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1, 121, 1, 2, 2, 4, 10, 3, 2, 2, 41, 1, 1, 1, 3, 7, 2, 2, 9, 4, 1, 3, 7, 6, 1, 1, 2, 2, 9, 3, 1, 1, 69, 4, 4, 5, 12, 1, 1, 5, 15, 1, 4

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..19999

BCMATH, Continued fraction expansion of the n-th root of a positive rational.

E. Bombieri and A. J. van der Poorten, Continued fractions of algebraic numbers, In: W. Bosma, A. van der Poorten (eds), Computational Algebra and Number Theory. Mathematics and Its Applications, vol. 325.

Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).

S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.

S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]

Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.

Eric Weisstein's World of Mathematics, Delian Constant.

G. Xiao, Contfrac

Index entries for continued fractions for constants

FORMULA

From Robert Israel, Jul 30 2014: (Start)

Bombieri/van der Poorten give a complicated formula:

a(n) = floor((-1)^(n+1)*3*p(n)^2/(q(n)*(p(n)^3-2*q(n)^3)) - q(n-1)/q(n)),

p(n+1) = a(n)*p(n) + p(n-1),

q(n+1) = a(n)*q(n) + q(n-1),

with a(1) = 1, p(1) = 1, q(1) = 0, p(2) = 1, q(2) = 1. (End)

EXAMPLE

2^(1/3) = 1.25992104989487316... = 1 + 1/(3 + 1/(1 + 1/(5 + 1/(1 + ...)))).

MAPLE

N:= 100: # to get a(1) to a(N)

a[1] := 1: p[1] := 1: q[1] := 0: p[2] := 1: q[2] := 1:

for n from 2 to N do

a[n] := floor((-1)^(n+1)*3*p[n]^2/(q[n]*(p[n]^3-2*q[n]^3)) - q[n-1]/q[n]);

p[n+1] := a[n]*p[n] + p[n-1];

q[n+1] := a[n]*q[n] + q[n-1];

od:

seq(a[i], i=1..N); # Robert Israel, Jul 30 2014

MATHEMATICA

ContinuedFraction[Power[2, (3)^-1], 70] (* Harvey P. Dale, Sep 29 2011 *)

PROG

(PARI) allocatemem(932245000); default(realprecision, 21000); x=contfrac(2^(1/3)); for (n=1, 20000, write("b002945.txt", n-1, " ", x[n])); \\ Harry J. Smith, May 08 2009

(Magma) ContinuedFraction(2^(1/3)); // Vincenzo Librandi, Oct 08 2017

CROSSREFS

Cf. A002946, A002947, A002948, A002949, A002580 (decimal expansion).

Cf. A002351, A002352 (convergents).

KEYWORD

cofr,nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

BCMATH link from Keith R Matthews (keithmatt(AT)gmail.com), Jun 04 2006

Offset changed by Andrew Howroyd, Jul 04 2024

STATUS

approved

page 1

Search completed in 0.010 seconds