A105609 Sylvester numbers for 1/(1+x^2).

1, 0, -1, -2, 1, -3, -1, 2, -1, 5, -1, 1, 1, -7, 1, 2, 1, -3, -1, 1, 1, -11, -1, 1, 1, 13, -1, 1, 1, 1, -1, 2, 1, 17, 1, 1, 1, -19, 1, 1, 1, 1, -1, 1, 1, -23, -1, 1, -1, 5, 1, 1, 1, -3, 1, 1, 1, 29, -1, 1, 1, -31, 1, 2, 1, 1, -1, 1, 1, 1, -1, 1, 1, 37

Offset: 0

Author

Paul Barry, Apr 15 2005

Keywords

• easy
• sign

Examples

			(x+I)(x-I)=1+x^2
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..2500
• Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
• Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.

Crossrefs

Cf. A020513, A105608, A105607.

Programs

Maple

A105609 := proc(n)local k: round(Re(mul(evalf(`if`(gcd(n+1, k)=1, I+I*exp(2*Pi*I*k/(n+1)), 1)),k=1..n))): end:
seq(A105609(n),n=0..20); # Nathaniel Johnston, Apr 20 2011
A105609 := proc(n) local k; mul(`if`(igcd(n+1,k)=1, 2*cos(Pi*k/(n+1)), 1), k=1..n) end; seq(round(A105609(n)), n = 0..73); # Peter Luschny, Jun 09 2011

Mathematica

f[n_] := FullSimplify[ Expand[Times @@ (I + I*Exp[2Pi*I*Select[Range[n], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 0, 32}] (* Robert G. Wilson v, Aug 02 2005 *)

Formula

a(n) = Product_{k=1..n} if(gcd(n+1, k)=1, (I+I*exp(2*Pi*I*k/(n+1))), 1), I=sqrt(-1).

alpha(n) = Product_{0A014963 with cos replaced by sin. - Peter Luschny, Jun 09 2011

Extensions

a(40)-a(73) from Nathaniel Johnston, Apr 20 2011