OFFSET
1,2
COMMENTS
D. H. Lehmer and E. Lehmer showed that the roots of these polynomials can be explicitly given, and that a(n) is divisible by 5^(n-1)*n^(2n-4).
The quotients a(n)/(5^(n-1)*n^(2n-4)) are 1, 1, 4, 81, 15625, 16777216, 137858491849, 7355827511386641, 2758702310349224820736, 7011372354671045074462890625, ...
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..46
D. H. Lehmer and E. Lehmer, Properties of polynomials having Fibonacci numbers for coefficients, Fibonacci Quarterly, Vol 21, No. 1 (1983), pp. 62-64.
FORMULA
a(n) ~ 5 * n^(2*n - 4) * phi^(2*n*(n-2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 02 2023
EXAMPLE
The first 5 polynomials are:
P_1(x) = 1
P_2(x) = x^2 + x - 1
P_3(x) = x^4 + x^3 + 2x^2 - x + 1
P_4(x) = x^6 + x^5 + 2x^4 + 3x^3 - 2x^2 + x - 1
P_5(x) = x^8 + x^7 + 2x^6 + 3x^5 + 5x^4 - 3x^3 + 2x^2 - x + 1
The discriminant of P_2(x), for example, is a(2) = 1^2 - 4*1*(-1) = 5.
MATHEMATICA
a={}; n=0; While[Length[a]<10, n++; f:=Fibonacci[Range[n]]; c = Join[Drop[Reverse[-(-1)^Range[n]]*f, -1], Reverse[f]]; p=x^Range[0, 2n-2].c; d=Discriminant[p, x]; AppendTo[a, d]]; a
PROG
(PARI) a(n) = if (n==1, 1, poldisc(sum(k=1, n, fibonacci(k)*x^(2*n-1-k)) + sum(k=1, n-1, (-1)^k*fibonacci(n-k)*x^(n-k-1)))); \\ Michel Marcus, Mar 02 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 30 2017
STATUS
approved