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

0,4

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.

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_{0<k<n} (if(gcd(k,n)=1, 2*cos(Pi*k/n), 1); a(n) = alpha(n+1). Cf. the formula in A014963 with cos replaced by sin. - Peter Luschny, Jun 09 2011

Example

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

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 *)

Crossrefs

Cf. A020513, A105608, A105607.

Sequence in context: A026100 A059127 A319494 * A268361 A330569 A340785

Adjacent sequences: A105606 A105607 A105608 * A105610 A105611 A105612

Keyword

easy,sign

Author

Paul Barry, Apr 15 2005

Extensions

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

Status

approved