A002351 - OEIS

%I M2380 N0945 #31 Jul 04 2024 20:07:42

%S 1,3,4,23,27,50,227,277,504,4309,4813,71691,76504,836731,1749966,

%T 2586697,12096754,147747745,307592244,1070524477,2448641198,

%U 3519165675,13006138223,55543718567,68549856790,124093575357,316737007504

%N Denominators of convergents to cube root of 2.

%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.

%D P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Robert Israel, <a href="/A002351/b002351.txt">Table of n, a(n) for n = 0..1975</a>

%H E. Bombieri and A. J. van der Poorten, <a href="https://doi.org/10.1007/978-94-017-1108-1_10">Continued fractions of algebraic numbers</a>, In: Bosma W., van der Poorten A. (eds) Computational Algebra and Number Theory. Mathematics and Its Applications, vol 325.

%H E. B. Burger, <a href="http://www.jstor.org/stable/2695737">Diophantine Olympics and World Champions: Polynomials and Primes Down Under</a>, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

%p Digits := 60: E := 2^(1/3); convert(evalf(E),confrac,50,'cvgts'): cvgts;

%p # Alternate:

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

%p c[0] := 1: p[0] := 1: a[0] := 0: p[1] := 1: a[1] := 1:

%p for n from 1 to N do

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

%p   p[n+1] := c[n]*p[n] + p[n-1];

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

%p od:

%p seq(a[i], i=1..N); # _Robert Israel_, Oct 08 2017

%t Denominator[Convergents[Surd[2,3],30]] (* _Harvey P. Dale_, Apr 02 2018 *)

%Y Cf. A002352 (numerators), A002945.

%K nonn,frac

%O 0,2

%A _N. J. A. Sloane_

%E Offset changed by _Andrew Howroyd_, Jul 04 2024