login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A284637 Discriminants of polynomials having Fibonacci numbers (A000045) for coefficients, P_n(x) = Sum_{k=1..n} F(k)*x^(2n-1-k) + Sum_{k=1..(n-1)} (-1)^k*F(n-k)*x^(n-k-1). 1
1, 5, 900, 2592000, 152587890625, 88060251340800000, 608462684559542896890625, 39491298245528363382865920000000, 24652445390187744298440793976121600000000, 136940866302168849110603332519531250000000000000000 (list; graph; refs; listen; history; text; internal format)
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.

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

CROSSREFS

Cf. A000045.

Sequence in context: A299837 A299726 A094630 * A173914 A015940 A093853

Adjacent sequences:  A284634 A284635 A284636 * A284638 A284639 A284640

KEYWORD

nonn

AUTHOR

Amiram Eldar, Mar 30 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 13 02:05 EDT 2020. Contains 336441 sequences. (Running on oeis4.)