A086823 Coefficient of x^2 in the n-th cyclotomic polynomial. (The same as the coefficient of x^(phi(n)-2) ).

0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, -1, 1, 1, 0, 0, 1, 0, 1, -1, 0, 1, 1, 0, 0, 1, 0, -1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, -1, 0, 1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, -1, 0, 0, 1, 0, 1, 1, 0, -1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1

Offset

1

Links

Antti Karttunen, Table of n, a(n) for n = 1..65539

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Formula

If n is odd, a(n) = 1/2 * mu(n)*(mu(n)-1), if n is even, a(n) = 1/2 * mu(n)*(mu(n)-1) - mu(n/2), where mu is Möbius mu function, A008683.

Mathematica

mm[n_]:=Module[{c=MoebiusMu[n]}, If[OddQ[n], (c(c-1))/2, (c(c-1))/2-MoebiusMu[ n/2]]]; Array[mm, 110] (* Harvey P. Dale, May 20 2018 *)

Prog

(PARI) a(n) = polcoeff(polcyclo(n), 2); \\ David Wasserman
(PARI) A086823(n) = (((1/2)*moebius(n)*(moebius(n)-1)) - if(!(n%2), moebius(n/2))); \\ Antti Karttunen, Sep 27 2018

Crossrefs

Cf. A008683, A000010.

Sequence in context: A373254 A152228 A368625 * A228487 A295306 A295303

Adjacent sequences: A086820 A086821 A086822 * A086824 A086825 A086826

Keyword

sign

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 07 2003

Extensions

More terms from David Wasserman, Mar 29 2005

Offset corrected by Antti Karttunen, Sep 27 2018

Status

approved