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 A186731 #39 May 10 2025 03:15:14
%S A186731 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,
%T A186731 10,11,22,11,12,24,12,13,26,13,14,28,14,15,30,15,16,32,16,17,34,17,18,
%U A186731 36,18,19,38,19,20,40,20,21,42,21,22,44,22,23,46,23,24,48
%N A186731 a(3n) = 2n, a(3n+1) = n, a(3n+2) = n+1.
%H A186731 Robert Israel, <a href="/A186731/b186731.txt">Table of n, a(n) for n = 0..10000</a>
%H A186731 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A186731 G.f.: (x*(1+x)/(1-x^3))^2.
%F A186731 a(n) = |A099254(n-2)| = |A099470(n-1)|. - _R. J. Mathar_, May 02 2013
%F A186731 From _Wesley Ivan Hurt_, Apr 28 2015: (Start)
%F A186731 a(n) = 2*a(n-3)-a(n-6).
%F A186731 a(n) = (n+1+n*0^mod(n,3)-mod(n+1,3))/3. (End)
%F A186731 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
%F A186731 From _Ridouane Oudra_, Nov 24 2024: (Start)
%F A186731 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.
%F A186731 a(n) = t(n+1)*t(n+3) - t(n-1)*t(n+1), where t(n) = A002264(n).
%F A186731 a(n) = A008130(n+1) - A008130(n-1). (End)
%F A186731 Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/2. - _Amiram Eldar_, May 10 2025
%p A186731 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):
%p A186731 map(f, [$0..100]); # _Robert Israel_, Apr 01 2016
%t A186731 CoefficientList[Series[(x*(1 + x)/(1 - x^3))^2, {x, 0, 100}], x] (* _Wesley Ivan Hurt_, Apr 28 2015 *)
%t A186731 LinearRecurrence[{0, 0, 2, 0, 0, -1}, {0, 0, 1, 2, 1, 2}, 100] (* _Vincenzo Librandi_, Apr 28 2015 *)
%o A186731 (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
%o A186731 (PARI) vector(50,n,n--;(n+1+n*0^(n%3)-(n+1)%3)/3) \\ _Derek Orr_, Apr 28 2015
%Y A186731 Column k = 2 of triangle in A198295.
%Y A186731 Cf. A000027, A001477, A005843, A011655, A185395, A185292.
%Y A186731 Cf. A002264, A008130.
%K A186731 easy,nonn
%O A186731 0,4
%A A186731 _Philippe Deléham_, Jan 21 2012
%E A186731 More terms from _Vincenzo Librandi_, Apr 28 2015