1,2

This continued fraction allows us to see that tan(1/sqrt(2)), sin(1/sqrt(2)), cos(1/sqrt(2)) are irrationnal. More generally, for any fixed positive integer m, the continued fraction for sqrt(m)*tan(1/sqrt(m)) is given by : a(12n-11)=a(12n-9)=a(12n-7)=a(12n-5)=a(12n-3)=a(12n-1)=1; a(12n-10)=12*m*n-9*m-2 ; a(12n-8)=12n-9 ; a(12n-6)=12*m*n-5*m-2 ; a(12n-4)=12n-5 ; a(12n-2)=12*m*n-m-2 ; a(12n)=12n-1

For n>=1 we have a(12n-11)=a(12n-9)=a(12n-7)=a(12n-5)=a(12n-3)=a(12n-1)=1; a(12n-10)=24n-20 ; a(12n-8)=12n-9 ; a(12n-6)=24n-12 ; a(12n-4)=12n-5 ; a(12n-2)=24n-4 ; a(12n)=12n-1

Cf. A123168.

Sequence in context: A010127 A055032 A039930 this_sequence A021246 A019633 A067277

Adjacent sequences: A123248 A123249 A123250 this_sequence A123252 A123253 A123254

nonn

Benoit Cloitre (abmt(AT)orange.fr), Oct 08 2006