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A087610
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Number of (-1,0,1) polynomials of degree-n irreducible over the integers.
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6
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3, 5, 12, 34, 104, 292, 916, 2791, 8660, 26538, 81584, 248554
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A087481 (irreducible polynomials of the form x^n +- x^(n-1) +- x^(n-2) +- ... +- 1), A087482 (irreducible binary polynomials).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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