A094754 Middle coefficient of cyclotomic polynomial Phi_n(x).

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

Offset

1,385

Comments

a(n) = 0 if n is a power of 2, otherwise a(n) is odd.

The first term with absolute value > 1 is a(385) = -3. - Michel Marcus, Apr 24 2019

Links

T. D. Noe, Table of n, a(n) for n=1..10000

Dorin Andrica, Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

G. P. Dresden, On the middle coefficient of a cyclotomic polynomial, Amer. Math. Monthly, 111 (No. 6, 2004), 531-533.

Gregory Dresden, On the Middle Coefficient of a Cyclotomic Polynomial, arXiv:1904.10593 [math.NT], 2019.

Maple

with(numtheory); t1:=[0, 0]; for n from 3 to 120 do t2:=cyclotomic(n, x); t3:=degree(t2, x); t1:=[op(t1), coeff(t2, x, floor(t3/2))]; od: t1;

Mathematica

Array[If[EvenQ@ Length@ #, 0, #[[Ceiling[Length[#]/2] ]] ] &@ CoefficientList[Cyclotomic[#, x], x] &, 105] (* Michael De Vlieger, Mar 29 2019 *)

Crossrefs

Cf. A095877 (n such that the middle coefficient of Phi_n(x) has a value not obtained for smaller n).

Sequence in context: A252744 A340373 A043545 * A321694 A262684 A287382

Adjacent sequences: A094751 A094752 A094753 * A094755 A094756 A094757

Keyword

sign

Author

N. J. A. Sloane, Jun 10 2004

Extensions

Second offset from Michel Marcus, Apr 24 2019

Status

approved