OFFSET
1,2
COMMENTS
a(3n - 1) = 2n (unsigned; n = 1, 2, 3...). In A099471, a(3n) = (2n + 1), unsigned. Odifreddi, p. 135: "Since the trefoil has polynomial x^2 - x + 1 and the quadrifoil (or flat knot) is the sum of two trefoils, its polynomial is (x^2 - x + 1)^2 = x^4 - 2x^3 + 3x^2 - 2x + 1."
Coefficient of x of the characteristic polynomial of the n X n matrix with 1's along the superdiagonal, main diagonal and subdiagonal. - John M. Campbell, Sep 14 2011
REFERENCES
Piergiorgio Odifreddi, "The Mathematical Century; The 30 Greatest Problems of the Last 100 Years", Princeton University Press, 2000, page 135.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).
FORMULA
M = the 4 X 4 companion matrix to the Quadrafoil polynomial x^4 - 2x^3 + 3x^2 - 2x + 1: [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 2 -3 2]. a(n) = rightmost term in M^n * [1 0 0 0].
O.g.f.: -x/(x^2-x+1)^2. a(n) = 2*a(n-1)-3*a(n-2)+2*a(n-3)-a(n-4) = -A099254(n-1). - R. J. Mathar, Apr 06 2008, Apr 23 2009
EXAMPLE
a(7) = -3 since M^7 * [1 0 0 0] = [2 4 2 -3].
MATHEMATICA
Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2+1] + KroneckerDelta[#1, #2-1] &, {n, n}], x], x], {n, 75}] (* John M. Campbell, Sep 14 2011 *)
Table[(3 n Cos[Pi n/3] - Sqrt[3] (3 n + 4) Sin[Pi n/3])/9, {n, 20}] (* Vladimir Reshetnikov, Sep 08 2016 *)
LinearRecurrence[{2, -3, 2, -1}, {-1, -2, -1, 2}, 90] (* Vincenzo Librandi, Sep 09 2016 *)
PROG
(Magma) I:=[-1, -2, -1, 2]; [n le 4 select I[n] else 2*Self(n-1)-3*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Sep 09 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Gary W. Adamson, Oct 17 2004
STATUS
approved