A087610 Number of (-1,0,1) polynomials of degree-n irreducible over the integers.

3, 5, 12, 34, 104, 292, 916, 2791, 8660, 26538, 81584, 248554

Offset: 1

Author

T. D. Noe, Sep 11 2003

Keywords

• nonn
• more

Comments

A (-1,0,1) polynomial is defined as a monic polynomial whose remaining coefficients are either -1, 0, or 1. For each n, there are 3^n polynomials to consider.

Examples

			a(2) = 5 because 1+x+x^2, 1+x^2, 1-x+x^2, -1+x+x^2, -1-x+x^2 are irreducible over the integers.
		

Links

• Eric Weisstein's World of Mathematics, Irreducible Polynomial

Crossrefs

Cf. A087481 (irreducible polynomials of the form x^n +- x^(n-1) +- x^(n-2) +- ... +- 1), A087482 (irreducible binary polynomials).

Programs

Maple

F:= proc(n) local T, count, t,x,p;
      if n::odd then
        T:= combinat:-cartprod([[-1,0,1]$(n-1),[1]])
      else
        T:= combinat:-cartprod([[-1,0,1]$(n-1),[-1,1]])
      fi;
      count:= 0;
      while not T[finished] do
         t:= T[nextvalue]();
         p:= x^n + add(t[i]*x^(n-i),i=1..n);
         if irreduc(p) then count:= count+1 fi;
      od;
      if n::odd then 2*count else count fi;
end proc:
3, seq(F(n),n=2..11); # Robert Israel, Dec 10 2015

Mathematica

Irreducible[p_, n_] := Module[{f}, f=FactorList[p, Modulus->n]; Length[f]==1 || Simplify[p-f[[2, 1]]]===0]; Table[xx=x^Range[0, n-1]; cnt=0; Do[p=x^n+xx.(IntegerDigits[i, 3, n]-1); If[Irreducible[p, 0], cnt++ ], {i, 0, 3^n-1}]; cnt, {n, 10}]

Extensions

a(11) and a(12) from Robert Israel, Dec 10 2015