OFFSET
1,1
COMMENTS
Conjecture: a(n) <= n^2+12 for all n>0.
Such a phenomenon happens quite often. In fact, for many interesting integer sequences a(k) (k=1,2,3,...), each of the polynomials x^n + sum_{k=0}^n a(k)*x^{n-k} (n>0) is irreducible modulo some prime not exceeding a*n^2+b*n+c, where a, b, c are suitable nonnegative constants.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..450
EXAMPLE
a(2) = 3 since x^2+x+2 is irreducible modulo 3 but reducible modulo 2.
Note also that a(13) = 181 = 13^2+12.
MATHEMATICA
A[n_, x_]:=A[n, x]=Sum[Fibonacci[k+1]*x^(n-k), {k, 0, n}]
Do[Do[If[IrreduciblePolynomialQ[A[n, x], Modulus->Prime[k]]==True, Print[n, " ", Prime[k]]; Goto[aa]], {k, 1, PrimePi[n^2+12]}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 07 2013
STATUS
approved