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%I #30 Feb 14 2023 08:58:17
%S 1,3,3,4,6,6,7,9,9,10,12,12,13,15,15,16,18,18,19,21,21,22,24,24,25,27,
%T 27,28,30,30,31,33,33,34,36,36,37,39,39,40,42,42,43,45,45,46,48,48,49,
%U 51,51,52,54,54,55,57,57,58,60,60,61,63,63,64,66,66,67,69,69,70,72,72
%N Expansion of (1+2x)/((1-x)(1-x^3)).
%C Partial sums of A010872(n+1).
%C Essentially the same as A130481. - _R. J. Mathar_, Jun 13 2008
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: (1+2x)/(1-x-x^3+x^4);
%F a(n) = 4/3 + n + 2*cos(Pi*2(n-1)/3)/3;
%F a(n) = Sum_{k=0..n} (k+1) mod 3;
%F a(n) = (n+1)*(n+2)/2 - 3*Sum_{k=0..n} floor((k+1)/3);
%F a(n) = 1 + n + Sum_{k=0..n} Jacobi(k, 3).
%F a(n) = a(n-1) + a(n-3) - a(n-4); a(0)=1, a(1)=3, a(2)=3, a(3)=4. - _Harvey P. Dale_, Sep 15 2011
%F a(n) = n + 1 when n + 2 is not a multiple of 3, and a(n) = n + 2 when n + 2 is a multiple of 3. - _Dennis P. Walsh_, Aug 06 2012
%F Sum_{n>=0} (-1)^n/a(n) = Pi/(3*sqrt(3)) + log(2)/3. - _Amiram Eldar_, Feb 14 2023
%p a:=n->add(chrem( [n,j], [1,3] ),j=1..n):seq(a(n), n=1..72);# _Zerinvary Lajos_, Apr 08 2009
%t f[n_]:=Mod[n,3];s=0;lst={};Do[AppendTo[lst,s+=f[n]],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2010 *)
%t CoefficientList[Series[(1+2x)/((1-x)(1-x^3)),{x,0,80}],x] (* or *) LinearRecurrence[{1,0,1,-1},{1,3,3,4},81] (* _Harvey P. Dale_, Sep 15 2011 *)
%Y Cf. A010872, A130481.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 24 2004
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