A099470 -id:A099470 - cp's OEIS frontend

A186731 a(3n) = 2n, a(3n+1) = n, a(3n+2) = n+1.

0, 0, 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, 21, 22, 44, 22, 23, 46, 23, 24, 48

Offset: 0

Philippe Deléham, Jan 21 2012

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Column k = 2 of triangle in A198295.
Cf. A000027, A001477, A005843, A011655, A185395, A185292.
Cf. A002264, A008130.

I:=[0,0,1,2,1,2]; [n le 6 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..80]]; // Vincenzo Librandi, Apr 28 2015

f:= gfun:-rectoproc({a(n)=2*a(n-3)-a(n-6), seq(a(i) = [0,0,1,2,1,2][i+1],i=0..5)},a(n),remember):
map(f, [$0..100]); # Robert Israel, Apr 01 2016

CoefficientList[Series[(x*(1 + x)/(1 - x^3))^2, {x, 0, 100}], x] (* Wesley Ivan Hurt, Apr 28 2015 *)
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {0, 0, 1, 2, 1, 2}, 100] (* Vincenzo Librandi, Apr 28 2015 *)

vector(50,n,n--;(n+1+n*0^(n%3)-(n+1)%3)/3) \\ Derek Orr, Apr 28 2015

G.f.: (x*(1+x)/(1-x^3))^2.
a(n) = |A099254(n-2)| = |A099470(n-1)|. - R. J. Mathar, May 02 2013
From Wesley Ivan Hurt, Apr 28 2015: (Start)
a(n) = 2*a(n-3)-a(n-6).
a(n) = (n+1+n*0^mod(n,3)-mod(n+1,3))/3. (End)
E.g.f.: (4/9)*x*exp(x) - (x/9)*exp(-x/2)*cos(sqrt(3)*x/2) - (sqrt(3)/9)*(2+x)*exp(-x/2)*sin(sqrt(3)*x/2). - Robert Israel, Apr 01 2016
From Ridouane Oudra, Nov 24 2024: (Start)
a(n) = n^3/6 - n/6 - (n^2 + 3*n/2 - 5/2)*floor(n/3) + (3*n/2 + 9/2)*floor(n/3)^2.
a(n) = t(n+1)*t(n+3) - t(n-1)*t(n+1), where t(n) = A002264(n).
a(n) = A008130(n+1) - A008130(n-1). (End)
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/2. - Amiram Eldar, May 10 2025

More terms from Vincenzo Librandi, Apr 28 2015

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

Original entry on oeis.org

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

Gary W. Adamson, Oct 17 2004

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."

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

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

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).

Cf. A099470.

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

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 *)

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)

cp's OEIS Frontend

A186731 a(3n) = 2n, a(3n+1) = n, a(3n+2) = n+1.

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