A002352 Numerators of convergents to cube root of 2. (Formerly M3260 N1316)

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

0,2

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

Cf. A002351 (denominators), A002945.

Sequence in context: A271602 A298827 A092659 * A042647 A266075 A041273

Adjacent sequences: A002349 A002350 A002351 * A002353 A002354 A002355

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

Offset changed by Andrew Howroyd, Jul 04 2024

Status

approved