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

Author

Philippe Deléham, Jan 21 2012

Keywords

• easy
• nonn

Links

• 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).

Crossrefs

Column k = 2 of triangle in A198295.

Cf. A000027, A001477, A005843, A011655, A185395, A185292.

Cf. A002264, A008130.

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Formula

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

Extensions

More terms from Vincenzo Librandi, Apr 28 2015