OFFSET
1,1
COMMENTS
Successive numbers n such that ChebyshevT[x, x/2] is not integer (is integer/2) see A007310
From Robert Israel, Apr 30 2015 (Start)
Numbers 2*k+1 where (1/2)*(2*k+1+2*sqrt(k*(k+1)))^(k+1/2)+(1/2)*(2*k+1+2*sqrt(k*(k+1)))^(-k-1/2) is not an integer.
All odd numbers not in A056220. (End)
From Robert Israel, May 01 2015 (Start)
ChebyshevT(1/2,n) = sqrt((n+1)/2) is an integer iff n is in A056220, otherwise it is irrational.
For odd k, ChebyshevT(k,x) = x*P(x^2) where P is a polynomial with integer coefficients and no roots >= 1.
Therefore if k is odd and n is a positive integer, ChebyshevT(k/2,n) = ChebyshevT(k,ChebyshevT(1/2,n)) = sqrt((n+1)/2)*P((n+1)/2) is an integer iff n is in A056220, otherwise it is irrational. (End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..1000
MAPLE
remove(t -> issqr((t+1)/2), [seq(2*j+1, j=0..1000)]); # Robert Israel, Apr 30 2015
MATHEMATICA
aa = {}; Do[If[PossibleZeroQ[Round[N[ChebyshevT[x/2, x], 100]] - N[ChebyshevT[x/2, x], 100]], , AppendTo[aa, x]], {x, 0, 1500}]; aa (* Artur Jasinski, Feb 14 2010 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Feb 14 2010
EXTENSIONS
Corrected and edited by Robert Israel, Apr 30 2015
STATUS
approved