A008133 a(n) = floor(n/3)*floor((n+1)/3).

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 9, 12, 16, 16, 20, 25, 25, 30, 36, 36, 42, 49, 49, 56, 64, 64, 72, 81, 81, 90, 100, 100, 110, 121, 121, 132, 144, 144, 156, 169, 169, 182, 196, 196, 210, 225, 225, 240, 256, 256, 272, 289, 289, 306, 324, 324, 342, 361, 361, 380, 400, 400, 420, 441, 441, 462, 484, 484, 506

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Oblong numbers and squares are subsequences: a(A016789(n)) = A002378(n); a(A008585(n)) = a(A016777(n)) = A000290(n). - Reinhard Zumkeller, Oct 09 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Vladimir Baltić, Applications of the finite state automata for counting restricted permutations and variations, Yugoslav Journal of Operations Research, 22 (2012), Number 2, 183-198; alternative link. - From _N. J. A. Sloane_, Jan 02 2013
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A000290, A008217, A008585, A016777, A016789, A087509.

Programs

Haskell

a008133 n = a008133_list !! n
a008133_list = zipWith (*) (tail ts) ts where ts = map (`div` 3) [0..]
-- Reinhard Zumkeller, Oct 09 2011

Magma

[Floor(n/3)*Floor((n+1)/3): n in [0..60]]; // Vincenzo Librandi, Aug 20 2011

Mathematica

Table[Floor[n/3]Floor[(n+1)/3],{n,0,100}] (* or *) LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,0,1,1,2,4},100] (* Harvey P. Dale, Sep 21 2024 *)

PARI

a(n) = floor(n/3)*floor((n+1)/3); /* Joerg Arndt, Mar 31 2013 */

Formula

From Paul Barry, Sep 14 2003: (Start)

Partial sums of A087509.

a(n+1) = Sum_{j=0..n} Sum_{k=0..j} [mod(j*k, 3)=2], where [] is the Iverson bracket. (End)

Empirical g.f.: -x^3*(x^2+1) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Mar 31 2013

From Amiram Eldar, May 10 2025: (Start)

Sum_{n>=3} 1/a(n) = Pi^2/3 + 1.

Sum_{n>=3} (-1)^(n+1)/a(n) = 2*log(2)-1. (End)