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

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

Author

Reinhard Zumkeller, May 18 2003

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A008130, A151842 (first differences).

Programs

Mathematica

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 *)

PARI

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

Formula

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