A094754 - OEIS

%I #19 Apr 25 2019 03:32:53

%S 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,

%T 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,

%U -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

%N Middle coefficient of cyclotomic polynomial Phi_n(x).

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

%C The first term with absolute value > 1 is a(385) = -3. - _Michel Marcus_, Apr 24 2019

%H T. D. Noe, <a href="/A094754/b094754.txt">Table of n, a(n) for n=1..10000</a>

%H Dorin Andrica, Ovidiu Bagdasar, <a href="http://hdl.handle.net/10545/623501">On some results concerning the polygonal polynomials</a>, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.

%H G. P. Dresden, <a href="http://www.jstor.org/stable/4145075">On the middle coefficient of a cyclotomic polynomial</a>, Amer. Math. Monthly, 111 (No. 6, 2004), 531-533.

%H Gregory Dresden, <a href="https://arxiv.org/abs/1904.10593">On the Middle Coefficient of a Cyclotomic Polynomial</a>, arXiv:1904.10593 [math.NT], 2019.

%p 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;

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

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

%K sign

%O 1,385

%A _N. J. A. Sloane_, Jun 10 2004

%E Second offset from _Michel Marcus_, Apr 24 2019