A163262 Denominators of fractions in the approximation of the square root of 2 by means of: f(n) = 3*f(n-1)/(f(n-1)^2+1); with f(1)= 1.

1, 2, 13, 493, 735853, 1619459312173, 7875984855578888541679213, 186030029004437379749629399827828117533654561726893

Offset: 1

Author

Mark Dols, Jul 23 2009

Keywords

• nonn
• frac

Comments

For root of c: f(n) = (1+c)*f(n-1)/(f(n-1)^2+1).

a(9) has 102 digits. - Emeric Deutsch, Jul 29 2009

Crossrefs

Cf. A002193 (sqrt(2)), A163261 (numerators).

Cf. A001601 and A051009: Newton's method.

Programs

Maple

f[1] := 1: for n from 2 to 10 do f[n] := 3*f[n-1]/(1+f[n-1]^2) end do: seq(denom(f[n]), n = 1 .. 8); # Emeric Deutsch, Jul 29 2009

PARI

f(n) = if (n==1, 1, 3*f(n-1)/(f(n-1)^2+1));
a(n) = denominator(f(n)); \\ Michel Marcus, Mar 04 2019

Extensions

a(7) and a(8) from Emeric Deutsch, Jul 29 2009

Name edited by Michel Marcus, Mar 04 2019