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%I #8 Mar 30 2012 18:36:32
%S 1,4,6,8,4,4,8,8,4,2,5,0,8,9,4,8,5,9,8,7,8,9,8,1,6,4,8,8,4,8,4,6,6,8,
%T 8,8,6,9,2,0,0,0,5,7,3,7,3,4,0,5,2,9,7,2,0,9,1,6,4,4,6,7,1,2,1,4,2,9,
%U 5,1
%N The limit of a continued fraction.
%C Consider the following Quet-like continued fraction recursion: x_0 = 1, x_1 = 1 + 1/x_1, x_2 = 1 + 1/(x_1 + 1/x_2), x_3 = 1 + 1/(x_1 + 1/(x_2 + 1/x_3)), ... x_n = 1 + 1/(x_1 + 1/(x_2 + 1/(x_3 + ... + 1/x_n))).
%D _Paul D. Hanna_, in a posting to sci.math news group entitled 'Limit of Continued Fraction Recursion' dated Jul 29 2002, 12:22.
%e Initial convergents are: x_0 = 1, x_1 = 1.618033988749895, x_2 = 1.431683416590579, x_3 = 1.477931798482607, x_4 = 1.466017958390778, x_5 = 1.469072006453889, x_6 = 1.468289031006081, x_7 = 1.468489818230137, x_8 = 1.468438335506229, x_9 = 1.468451536645148, x_10 = 1.46844815169046, ...
%p f := proc(m, n) local u, x, expr, k; expr := x; u := 1; for k to n do expr := simplify(subs(x = u + 1/x, expr)); u := fsolve(x = expr, x = 0 .. 2); if m < k then print(k, u) end if end do end proc
%Y The partial quotients are in A072981.
%K nonn,cons
%O 1,2
%A _Paul D. Hanna_, _Robert G. Wilson v_ and Denis Feldmann (denis.feldmann(AT)wanadoo.fr), Jul 29 2002
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