A092200 Expansion of (1+2x)/((1-x)(1-x^3)).

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, 27, 28, 30, 30, 31, 33, 33, 34, 36, 36, 37, 39, 39, 40, 42, 42, 43, 45, 45, 46, 48, 48, 49, 51, 51, 52, 54, 54, 55, 57, 57, 58, 60, 60, 61, 63, 63, 64, 66, 66, 67, 69, 69, 70, 72, 72

Offset: 0

Author

Paul Barry, Feb 24 2004

Keywords

• easy
• nonn

Comments

Partial sums of A010872(n+1).

Essentially the same as A130481. - R. J. Mathar, Jun 13 2008

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A010872, A130481.

Programs

Maple

a:=n->add(chrem( [n,j], [1,3] ),j=1..n):seq(a(n), n=1..72);# Zerinvary Lajos, Apr 08 2009

Mathematica

f[n_]:=Mod[n,3];s=0;lst={};Do[AppendTo[lst,s+=f[n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
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 *)

Formula

G.f.: (1+2x)/(1-x-x^3+x^4);

a(n) = 4/3 + n + 2*cos(Pi*2(n-1)/3)/3;

a(n) = Sum_{k=0..n} (k+1) mod 3;

a(n) = (n+1)*(n+2)/2 - 3*Sum_{k=0..n} floor((k+1)/3);

a(n) = 1 + n + Sum_{k=0..n} Jacobi(k, 3).

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

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

Sum_{n>=0} (-1)^n/a(n) = Pi/(3*sqrt(3)) + log(2)/3. - Amiram Eldar, Feb 14 2023