OFFSET
0,3
COMMENTS
See a May 06 2017 comment on A049310 where these problems are considered which originated in a conjecture by Michel Lagneau (see A008611) on Fibonacci polynomials.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
FORMULA
a(n) = 2*b(n) if n is even and 1 + 2*b(n) if n is odd with b(n) = floor(n/2) - floor((n+1)/6) = A286716(n). See the g.f. for {b(n)}_{n>=0} there.
From Colin Barker, May 18 2017: (Start)
G.f.: x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
(End)
EXAMPLE
a(4) = 2: S(4, x) = 1+x^4-3*x^2, and only two of the four zeros -phi, -1/phi, +1/phi, phi are in the open interval (-phi, +phi), the other two are at the borders.
MATHEMATICA
CoefficientList[Series[x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)), {x, 0, 50}], x] (* G. C. Greubel, Mar 08 2018 *)
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 2, 3}, 80] (* Harvey P. Dale, Aug 20 2020 *)
PROG
(PARI) concat(0, Vec(x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ Colin Barker, May 18 2017
(Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)))); // G. C. Greubel, Mar 08 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, May 13 2017
STATUS
approved