OFFSET
0,2
REFERENCES
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Robert Israel, Table of n, a(n) for n = 0..1975
E. Bombieri and A. J. van der Poorten, Continued fractions of algebraic numbers, In: Bosma W., van der Poorten A. (eds) Computational Algebra and Number Theory. Mathematics and Its Applications, vol 325.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
FORMULA
From Robert Israel, Oct 08 2017: (Start)
c(n) = floor((-1)^n*3*a(n)^2/(q(n)*(a(n)^3-2*q(n)^3)) - q(n-1)/q(n)),
a(n+1) = c(n)*a(n) + a(n-1),
q(n+1) = c(n)*q(n) + q(n-1), with a(0) = 1, c(0) = 1, q(0) = 0, a(1) = 1, q(1) = 1. (End)
MAPLE
Digits := 60: E := 2^(1/3); convert(evalf(E), confrac, 50, 'cvgts'): cvgts;
# Alternate:
N:= 100: # to get a(1) to a(N)
c[0] := 1: a[0] := 1: q[0] := 0: a[1] := 1: q[1] := 1:
for n from 1 to N do
c[n] := floor((-1)^n*3*a[n]^2/(q[n]*(a[n]^3-2*q[n]^3)) - q[n-1]/q[n]);
a[n+1] := c[n]*a[n] + a[n-1];
q[n+1] := c[n]*q[n] + q[n-1];
od: seq(a[i], i=1..N); # Robert Israel, Oct 08 2017
MATHEMATICA
Convergents[CubeRoot[2], 30]//Numerator (* Harvey P. Dale, May 30 2023 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
EXTENSIONS
Offset changed by Andrew Howroyd, Jul 04 2024
STATUS
approved