%I #20 Oct 11 2021 18:45:03

%S 2,4,24,336,20832,15290688,648294589824,19853227004192875776,

%T 25742087295468908558449985705472,

%U 1022127038655087543318858397597063429079854268617728

%N 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.

%C 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

%F From _Jianing Song_, Oct 10 2021: (Start)

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

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

%F 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)

%e The n-th application of the mapping produces the fraction x_n/y_n from the fraction x_(n-1)/y_(n-1):

%e n=1: f(1/2) = (1+2)/(2*1*2) = 3/4 (so a(1)=4);

%e n=2: f(3/4) = (3+4)/(2*3*4) = 7/24 (so a(2)=24);

%e n=3: f(7/24) = (7+24)/(2*7*24) = 31/336 (so a(3)=336).

%e From _Jianing Song_, Oct 10 2021: (Start)

%e a(0) = 2;

%e a(1) = 2 * 2 * 1 = 4;

%e a(2) = 2 * 4 * (1+2) = 24;

%e a(3) = 2 * 24 * (1+2+4) = 336;

%e a(4) = 2 * 336 * (1+2+4+24) = 20832. (End)

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

%o (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

%Y Cf. A081475 (numerators).

%K nonn

%O 0,1

%A _Amarnath Murthy_, Mar 24 2003

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

%E Rational numbers listed in Name corrected by _Jon E. Schoenfield_, Apr 24 2014