A095878 Value of middle coefficient of the cyclotomic polynomial Phi_n(x) for n in A095877.

1, 0, -1, -3, 3, 5, -5, -7, 7, 19, -11, 25, 11, 27, -9, 9, 45, 13, -35, -15, -23, 17, 349, -65, -27, -17, 15, -13, 21, -21, -19, 165, -29, 23, -37, 91, 29, 37, 33, -63, 161, 69, 41

Offset

1,4

Comments

After the second term, the entries are always odd; see Dresden's article for proof.

Links

Table of n, a(n) for n=1..43.

G. Dresden, On the Middle Coefficient of a Cyclotomic Polynomial (MAA Monthly, 2004, 531-533).

Maple

vals:= {1, 0, -1}:
a[1]:= 1; a[2]:= 0: a[3]:= -1:
count:= 3:
for n from 4 while count < 20 do
  f:= ifactors(n)[2];
  if nops(f) <= 2 or max(f[.., 2])>1 then next fi;
  p:= numtheory:-cyclotomic(n, x);
  c:= coeff(p, x, degree(p, x)/2);
  if not member(c, vals) then
    count:= count+1;
    a[count]:= c;
    vals:= vals union {c};
  fi
od:
seq(a[i], i=1..count); # Robert Israel, Dec 18 2014

Mathematica

vals={1, 0, -1}; lst={3, 4, 6}; Do[pwrs=Transpose[FactorInteger[n]][[2]]; If[Length[pwrs]>2 && 1==Times@@pwrs, c=Coefficient[Cyclotomic[n, x], x, EulerPhi[n]/2]; If[ !MemberQ[vals, c], AppendTo[lst, n]; AppendTo[vals, c]]], {n, 7, 50000, 2}]; vals

Crossrefs

Cf. A094754 (middle coefficient of cyclotomic polynomial Phi_n(x)).

Sequence in context: A333153 A245146 A339102 * A341143 A369337 A157966

Adjacent sequences: A095875 A095876 A095877 * A095879 A095880 A095881

Keyword

hard,sign

Author

T. D. Noe, Jun 10 2004

Extensions

More terms from Jozsef Pelikan, Dec 18 2014

Status

approved