A099471 A sequence generated from the Quadrifoil (flat knot).

1, 0, -2, -3, -1, 3, 5, 2, -4, -7, -3, 5, 9, 4, -6, -11, -5, 7, 13, 6, -8, -15, -7, 9, 17, 8, -10, -19, -9, 11, 21, 10, -12, -23, -11, 13, 25, 12, -14, -27, -13, 15, 29, 14, -16, -31, -15, 17, 33, 16, -18, -35, -17, 19, 37, 18, -20, -39, -19, 21, 41, 20, -22

Offset: 0

Author

Gary W. Adamson, Oct 17 2004

Keywords

• sign
• easy

Comments

a(3*n), n = 1,2,3... = 2*n + 1, unsigned. Odifreddi, p. 135 states: "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) = x^4 - 2*x^3 + 3*x^2 - 2*x + 1."

Examples

			a(6) = 5 since M^6 * [1 1 1 1] = [ -3 -1 3 5].
		

References

• P. Odifreddi, "The Mathematical Century; The 30 Greatest Problems of the Last 100 Years", Princeton University Press, page 135.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Crossrefs

Cf. A099470.

Programs

Maple

a:= proc(n) local m, r; r:= 1+irem(n, 6, 'm');
      [1, 0, -2, -3, -1, 3][r] +m*[4, 2, -2, -4, -2, 2][r]
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 25 2013

Mathematica

Table[((9 + 6 n) Cos[Pi n/3] - 5 Sqrt[3] Sin[Pi n/3])/9, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 09 2016 *)
LinearRecurrence[{2,-3,2,-1},{1,0,-2,-3},70] (* Harvey P. Dale, Mar 27 2025 *)

Formula

a(n) = M^n * [1 1 1 1], rightmost term; where M = the 4 X 4 companion matrix to the Quadrifoil polynomial x^4 - 2*x^3 + 3*x^2 - 2*x + 1: [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / -1 2 -3 2].

G.f.: -(x^3-x^2+2*x-1) / (x^2-x+1)^2. - Colin Barker, May 25 2013

a(n+1) = 1 - sum(A101950(n-k+2, k+2), k=0..floor(n/2)) - Johannes W. Meijer, Aug 06 2013

From A.H.M. Smeets, Sep 13 2018 (Start)

a(3*k) = a(3*k-1) + a(3*k+1) for k > 0.

a(3*k) = (-1)^k*(2*k+1) for k >= 0.

a(3*k+1) = (-1)^k*k for k >= 0.

a(3*k+2) = (-1)^(k+1)*(k+2) for k >= 0. (End)