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

0,2

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

Links

Table of n, a(n) for n=0..10.

Formula

From Jianing Song, Oct 10 2021: (Start)

a(n) = a(n-1) + A081476(n-1) for n >= 1 with a(0) = 1 and A081476(0) = 2.

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

Cf. A001685, A003686, A064526.

The denominators are A081476.

Sequence in context: A156895 A074047 A121810 * A123212 A213437 A070231

Adjacent sequences: A081472 A081473 A081474 * A081476 A081477 A081478

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