(* Program to show the relations
of El'ahir part1 *)
acc:=300
(* we should give enough accuracy (acc), in order to
compute the rationalize[one] term *)
a=13 ;
b=7 ;
k:=a/b
one:=N[Cos[n*ArcCos[k]] , acc]
one1:=Rationalize[one]
two:=Sum[((-1)^r)*((2*k)^(n-2*r))*(n/(n-r))*
Binomial[n-r,r] , {r,0,Floor[n/2]}]
ask1:=TrueQ[2*one1==two]
ask2:=IntegerQ[(b^n)*one1]
ask3:=TrueQ[ask1 && ask2]
Table[ask3 , {n,1,99}]
{True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True, True, True, True, True, True,
True, True, True}