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A002352 Numerators of convergents to cube root of 2.

%I A002352 M3260 N1316 #32 May 27 2025 14:25:14
%S A002352 1,4,5,29,34,63,286,349,635,5429,6064,90325,96389,1054215,2204819,
%T A002352 3259034,15240955,186150494,387541943,1348776323,3085094589,
%U A002352 4433870912,16386707325,69980700212,86367407537,156348107749,399063623035,5743238830239,17628780113752
%N A002352 Numerators of convergents to cube root of 2.
%D A002352 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
%D A002352 P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
%D A002352 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002352 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002352 Robert Israel, <a href="/A002352/b002352.txt">Table of n, a(n) for n = 0..1975</a>
%H A002352 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 A002352 E. B. Burger, <a href="https://web.archive.org/web/20240529063505/http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Burger.pdf">Diophantine Olympics and World Champions: Polynomials and Primes Down Under</a>, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
%F A002352 From _Robert Israel_, Oct 08 2017: (Start)
%F A002352 c(n) = floor((-1)^n*3*a(n)^2/(q(n)*(a(n)^3-2*q(n)^3)) - q(n-1)/q(n)),
%F A002352 a(n+1) = c(n)*a(n) + a(n-1),
%F A002352 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)
%p A002352 Digits := 60: E := 2^(1/3); convert(evalf(E),confrac,50,'cvgts'): cvgts;
%p A002352 # Alternate:
%p A002352 N:= 100: # to get a(1) to a(N)
%p A002352 c[0] := 1: a[0] := 1: q[0] := 0: a[1] := 1: q[1] := 1:
%p A002352 for n from 1 to N do
%p A002352   c[n] := floor((-1)^n*3*a[n]^2/(q[n]*(a[n]^3-2*q[n]^3)) - q[n-1]/q[n]);
%p A002352   a[n+1] := c[n]*a[n] + a[n-1];
%p A002352   q[n+1] := c[n]*q[n] + q[n-1];
%p A002352 od: seq(a[i], i=1..N); # _Robert Israel_, Oct 08 2017
%t A002352 Convergents[CubeRoot[2],30]//Numerator (* _Harvey P. Dale_, May 30 2023 *)
%Y A002352 Cf. A002351 (denominators), A002945.
%K A002352 nonn,frac
%O A002352 0,2
%A A002352 _N. J. A. Sloane_
%E A002352 Offset changed by _Andrew Howroyd_, Jul 04 2024