A008130 -id:A008130 - 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

A082667 a(n) = floor(2n/3) * ceiling(2n/3) / 2.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 10, 15, 18, 21, 28, 32, 36, 45, 50, 55, 66, 72, 78, 91, 98, 105, 120, 128, 136, 153, 162, 171, 190, 200, 210, 231, 242, 253, 276, 288, 300, 325, 338, 351, 378, 392, 406, 435, 450, 465, 496, 512, 528, 561, 578, 595, 630, 648, 666, 703, 722, 741

Offset: 1

Reinhard Zumkeller, May 18 2003

Prefixing with 0,0,0 gives the sequence c(n) defined as the number of (x,y,z) satisfying 2w = 3x-3y where w,x,y are all in {1,...,n}, for n>=0; see the Formula section.

For n >= 2, numbers k such that floor(sqrt(2k)+1/2) | 2k. - Wesley Ivan Hurt, Dec 01 2020

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

Cf. A008130, A151842 (first differences).

n2[n_]:=Module[{c=2*n/3},(Floor[c]Ceiling[c])/2]; Array[n2,60] (* Harvey P. Dale, Feb 03 2012 *)
LinearRecurrence[{1,0,2,-2,0,-1,1},{0,1,2,3,6,8,10},60] (* Robert G. Wilson v, Jun 06 2014 *)

a(n) = (2*n\3) * ceil(2*n/3) / 2; \\ Amiram Eldar, May 10 2025

a(n) = a(n-1) + 2a(n-3) - 2a(n-4) - a(n-6) + a(n-7), (with 0,0,0 prefixed as in the Comments section). - Clark Kimberling, Apr 15 2012
a(n) = floor((n + 1)/3)*(n - floor((n + 1)/3)). - Wesley Ivan Hurt, Jun 06 2014
G.f.: -x^2*(1+x)*(1+x^2) / ( (1+x+x^2)^2*(x-1)^3 ). - R. J. Mathar, Jun 07 2014
From Amiram Eldar, May 10 2025: (Start)
Sum_{n>=2} 1/a(n) = 2 + Pi^2/12.
Sum_{n>=2} (-1)^n/a(n) = Pi - Pi^2/24 - 2. (End)
E.g.f.: exp(-x/2)*(2*exp(3*x/2)*(3*x^2 + 3*x - 1) - (3*x - 2)*cos(sqrt(3)*x/2) + sqrt(3)*x*sin(sqrt(3)*x/2))/27. - Stefano Spezia, May 11 2025

cp's OEIS Frontend

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

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