This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008133 #50 May 10 2025 03:16:01
%S A008133 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,
%T A008133 72,81,81,90,100,100,110,121,121,132,144,144,156,169,169,182,196,196,
%U A008133 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
%N A008133 a(n) = floor(n/3)*floor((n+1)/3).
%C A008133 Oblong numbers and squares are subsequences: a(A016789(n)) = A002378(n); a(A008585(n)) = a(A016777(n)) = A000290(n). - _Reinhard Zumkeller_, Oct 09 2011
%H A008133 Vincenzo Librandi, <a href="/A008133/b008133.txt">Table of n, a(n) for n = 0..10000</a>
%H A008133 Vladimir Baltić, <a href="https://doi.org/10.2298/YJOR120211023B">Applications of the finite state automata for counting restricted permutations and variations</a>, Yugoslav Journal of Operations Research, 22 (2012), Number 2, 183-198; <a href="http://yujor.fon.bg.ac.rs/index.php/yujor/article/view/395">alternative link</a>. - From _N. J. A. Sloane_, Jan 02 2013
%H A008133 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A008133 From _Paul Barry_, Sep 14 2003: (Start)
%F A008133 Partial sums of A087509.
%F A008133 a(n+1) = Sum_{j=0..n} Sum_{k=0..j} [mod(j*k, 3)=2], where [] is the Iverson bracket. (End)
%F A008133 Empirical g.f.: -x^3*(x^2+1) / ((x-1)^3*(x^2+x+1)^2). - _Colin Barker_, Mar 31 2013
%F A008133 From _Amiram Eldar_, May 10 2025: (Start)
%F A008133 Sum_{n>=3} 1/a(n) = Pi^2/3 + 1.
%F A008133 Sum_{n>=3} (-1)^(n+1)/a(n) = 2*log(2)-1. (End)
%t A008133 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 *)
%o A008133 (Magma) [Floor(n/3)*Floor((n+1)/3): n in [0..60]]; // _Vincenzo Librandi_, Aug 20 2011
%o A008133 (Haskell)
%o A008133 a008133 n = a008133_list !! n
%o A008133 a008133_list = zipWith (*) (tail ts) ts where ts = map (`div` 3) [0..]
%o A008133 -- _Reinhard Zumkeller_, Oct 09 2011
%o A008133 (PARI) a(n) = floor(n/3)*floor((n+1)/3); /* _Joerg Arndt_, Mar 31 2013 */
%Y A008133 Cf. A000290, A008217, A008585, A016777, A016789, A087509.
%K A008133 nonn,easy
%O A008133 0,6
%A A008133 _N. J. A. Sloane_