A081478 Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the denominators.

1, 2, 2, -2, 6, -6, 42, -42, 1806, -1806, 3263442, -3263442, 10650056950806, -10650056950806, 113423713055421844361000442, -113423713055421844361000442, 12864938683278671740537145998360961546653259485195806

Offset: 1

Author

Amarnath Murthy, Mar 24 2003

Keywords

• sign
• frac

Comments

The mapping f(a/b) = (a + b)/(a - b). Taking a = 2 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the periodic sequence 2/1,3/1,2/1,3/1,...

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..26

Crossrefs

A003687 gives the numerators.

Cf. A007018.

Programs

Mathematica

Last /@ NestList[{(#1 - #2), #1 #2} & @@ # &, {2, 1}, 16] (* Michael De Vlieger, Sep 04 2016 *)

Sage

# Variant with first four terms slightly different. Absolute values.
def A081478_abs():
    x, y = 1, 2
    yield x
    while True:
       yield x
       x, y = x * y, x//y + 1
a = A081478_abs(); print([next(a) for i in range(17)])  # Peter Luschny, Dec 17 2015

Formula

a(2n-1) = A007018(n-1), a(2n) = -A007018(n-1) for n >= 2. - Jianing Song, Oct 10 2021

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003