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A105609
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Sylvester numbers for 1/(1+x^2).
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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
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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).
Contribution from Peter Luschny, Jun 9 2011: (Start) 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. (End)
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EXAMPLE
| (x+I)(x-I)=1+x^2
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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 9 2011
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MATHEMATICA
| f[n_] := FullSimplify[ Expand[Times @@ (I + I*Exp[2Pi*I*Select[Range[n], GCD[ #, n] == 1 &]/n])]]; Table[ f[n], {n, 0, 32}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 02 2005)
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CROSSREFS
| Cf. A020513, A105608, A105607.
Sequence in context: A065704 A026100 A059127 * A206778 A101872 A174892
Adjacent sequences: A105606 A105607 A105608 * A105610 A105611 A105612
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KEYWORD
| easy,sign
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Apr 15 2005
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EXTENSIONS
| a(40) - a(73) from Nathaniel Johnston (nathaniel(AT)nathanieljohnston.com), Apr 20 2011
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