A081475 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 numerator.
1, 3, 7, 31, 367, 21199, 15311887, 648309901711, 19853227652502777487, 25742087295488761786102488482959, 1022127038655087543344600484892552190865956757100687
Offset: 0
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)=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)
Programs
-
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
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)
Extensions
Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
Edited by Jon E. Schoenfield, Apr 25 2014
Comments