A002352 Numerators of convergents to cube root of 2.

1, 4, 5, 29, 34, 63, 286, 349, 635, 5429, 6064, 90325, 96389, 1054215, 2204819, 3259034, 15240955, 186150494, 387541943, 1348776323, 3085094589, 4433870912, 16386707325, 69980700212, 86367407537, 156348107749, 399063623035, 5743238830239, 17628780113752

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

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.

Crossrefs

Cf. A002351 (denominators), A002945.

Programs

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 *)

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)

Extensions

Offset changed by Andrew Howroyd, Jul 04 2024