OFFSET
0,2
COMMENTS
An infinite coprime sequence defined by recursion.
Every term is relatively prime to all others. - Michael Somos, Feb 01 2004
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
FORMULA
From Jianing Song, Oct 10 2021: (Start)
a(0) = 1, a(n) = a(n-1) + 2^n*a(0)*a(1)*...*a(n-2) for n >= 1.
a(0) = 1, a(1) = 3, a(n) = a(n-1) + 2*a(n-2)*(a(n-1)-a(n-2)) for n >= 2. (End)
EXAMPLE
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)=3);
n=2: f(3/4) = (3+4)/(2*3*4) = 7/24 (so a(2)=7);
n=3: f(7/24) = (7+24)/(2*7*24) = 31/336 (so a(3)=31).
From Jianing Song, Oct 10 2021: (Start)
a(0) = 1;
a(1) = 1 + 2^1 = 3;
a(2) = 3 + 2^2*1 = 7;
a(3) = 7 + 2^3*1*3 = 31;
a(4) = 31 + 2^4*1*3*7 = 367;
a(5) = 367 + 2^5*1*3*7*31 = 21199. (End)
PROG
(PARI) a(n)=local(v); if(n<2, n>0, v=[1, 2]; for(k=2, n, v=[v[1]+v[2], 2*v[1]*v[2]]); v[1])
(PARI) lista(n) = my(v=vector(n+1)); v[1]=1; if(n>=1, v[2]=3); for(k=2, n, v[k+1] = v[k] + 2*v[k-1]*(v[k]-v[k-1])); v \\ Jianing Song, Oct 10 2021
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Amarnath Murthy, Mar 24 2003
EXTENSIONS
Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
Edited by Jon E. Schoenfield, Apr 25 2014
STATUS
approved