A081476 Consider the mapping f(x/y) = (x+y)/(2xy) where x/y is a reduced fraction. Beginning with x_0 = 1 and y_0 = 2, repeated application of this mapping produces a sequence of fractions x_n/y_n; a(n) is the n-th denominator.

2, 4, 24, 336, 20832, 15290688, 648294589824, 19853227004192875776, 25742087295468908558449985705472, 1022127038655087543318858397597063429079854268617728

Offset: 0

Author

Amarnath Murthy, Mar 24 2003

Keywords

• nonn

Comments

Note that gcd(x+y,2*x*y) <= gcd(x+y,2)*gcd(x+y,x)*gcd(x+y,y), so gcd(x,y) = 1 implies gcd(x+y,2*x*y) = 1 unless both x,y are odd. As a result, the definition gives x_{n+1} = x_n+y_n and y_{n+1} = 2*(x_n)*(y_n) with x_0 = 1 and y_0 = 2. - Jianing Song, Oct 10 2021

Examples

			The n-th application of the mapping produces the fraction x_n/y_n from the fraction x_(n-1)/y_(n-1):
n=1:  f(1/2) = (1+2)/(2*1*2) = 3/4 (so a(1)=4);
n=2:  f(3/4) = (3+4)/(2*3*4) = 7/24 (so a(2)=24);
n=3:  f(7/24) = (7+24)/(2*7*24) = 31/336 (so a(3)=336).
From _Jianing Song_, Oct 10 2021: (Start)
a(0) = 2;
a(1) = 2 * 2 * 1 = 4;
a(2) = 2 * 4 * (1+2) = 24;
a(3) = 2 * 24 * (1+2+4) = 336;
a(4) = 2 * 336 * (1+2+4+24) = 20832. (End)
		

Crossrefs

Cf. A081475 (numerators).

Programs

Mathematica

nxt[{a_,b_}]:={(a+b),2a b}; Transpose[NestList[nxt,{1,2},10]][[2]] (* Harvey P. Dale, Jan 13 2012 *)

PARI

lista(n) = my(v=vector(n+1)); v[1]=2; if(n>=1, v[2]=4); for(k=2, n, v[k+1] = 2*v[k]*v[k-1] + v[k]^2/v[k-1]); v \\ Jianing Song, Oct 10 2021

Formula

From Jianing Song, Oct 10 2021: (Start)

a(n) = 2 * A081475(n-1) * a(n-1) for n >= 1 with A081475(0) = 1 and a(0) = 2.

a(0) = 2, a(n) = 2 * a(n-1) * (1+a(0)+a(1)+...+a(n-2)) for n >= 1.

a(0) = 2, a(1) = 4, a(n) = 2*a(n-1)*a(n-2) + (a(n-1))^2/a(n-2) for n >= 2. (End)

Extensions

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

Rational numbers listed in Name corrected by Jon E. Schoenfield, Apr 24 2014