A099470 A sequence generated from the Quadrifoil.

-1, -2, -1, 2, 4, 2, -3, -6, -3, 4, 8, 4, -5, -10, -5, 6, 12, 6, -7, -14, -7, 8, 16, 8, -9, -18, -9, 10, 20, 10, -11, -22, -11, 12, 24, 12, -13, -26, -13, 14, 28, 14, -15, -30, -15, 16, 32, 16, -17, -34, -17, 18, 36, 18, -19, -38, -19, 20, 40, 20, -21, -42

Offset: 1

Author

Gary W. Adamson, Oct 17 2004

Keywords

• sign
• easy

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

Examples

			a(7) = -3 since M^7 * [1 0 0 0] = [2 4 2 -3].
		

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).

Crossrefs

Cf. A099254, A099471.

Programs

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

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 *)

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