A071851 The limit of a continued fraction.

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, 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, 5, 1

Offset

1,2

Comments

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))).

References

Paul D. Hanna, in a posting to sci.math news group entitled 'Limit of Continued Fraction Recursion' dated Jul 29 2002, 12:22.

Links

Table of n, a(n) for n=1..70.

Example

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, ...

Maple

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

Crossrefs

The partial quotients are in A072981.

Sequence in context: A201520 A179022 A021685 * A083256 A083257 A286366

Adjacent sequences: A071848 A071849 A071850 * A071852 A071853 A071854

Keyword

nonn,cons

Author

Paul D. Hanna, Robert G. Wilson v and Denis Feldmann (denis.feldmann(AT)wanadoo.fr), Jul 29 2002

Status

approved