OFFSET
1,1
COMMENTS
The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(8421,x) =1 + x^2 + x^5 + x^6 + x^7 + x^13
= (1 + x) (1 + x + x^2)*f(x), where
f(x) = 1 - 2 x + 3 x^2 - 3 x^3 + 2 x^4 - x^7 + 2 x^8 - 2 x^9 + x^10.
This show that a factor of p(8421,x) has a factor that has -3 as a coefficient. Actually, 8421 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2,3}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208181 are disjoint.
MATHEMATICA
t = Table[IntegerDigits[n, 2], {n, 1, 25000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x], Factor[p[n, x]]}, {n, 1, 25000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {___, -3, ___}]} &,
Map[{#[[1]], CoefficientList[#[[2]], x]} &,
Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
Table[{n, Factor[p[n, x]]}, {n, 1, 24900}]]]]], {_, {}}]
Map[#[[1]] &, %] (* A208182 *)
(* Peter J. C. Moses, Feb 22 1012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 24 2012
STATUS
approved