A117910 Expansion of (1 + x + x^2 + x^4)/((1-x^3)*(1-x^6)).

1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 2, 3, 5, 3, 3, 6, 3, 4, 7, 4, 4, 8, 4, 5, 9, 5, 5, 10, 5, 6, 11, 6, 6, 12, 6, 7, 13, 7, 7, 14, 7, 8, 15, 8, 8, 16, 8, 9, 17, 9, 9, 18, 9, 10, 19, 10, 10, 20, 10, 11, 21, 11, 11, 22, 11, 12, 23, 12, 12, 24, 12, 13, 25, 13, 13, 26, 13, 14, 27, 14

Offset: 0

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• nonn

Comments

Diagonal sums of A117908.

Appears to be a permutation of floor((n+5)/5).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Crossrefs

Cf. A002264, A014125, A024164, A144677, A117908, A152467, A175676.

Programs

Magma

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x+x^2+x^4)/((1-x^3)*(1-x^6)) )); // G. C. Greubel, Oct 21 2021

Mathematica

CoefficientList[Series[(1+x+x^2+x^4)/((1-x^3)(1-x^6)),{x,0,100}],x] (* or *) LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{1,1,1,1,2,1,2,3,2},100] (* Harvey P. Dale, Apr 10 2014 *)
Table[If[Mod[n,3]==1, Mod[Binomial[n+2,3], n+2], Floor[(n+6)/6]], {n, 0, 100}] (* G. C. Greubel, Nov 18 2021 *)

Sage

def A117910_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x+x^2+x^4)/((1-x^3)*(1-x^6)) ).list()
A117910_list(100) # G. C. Greubel, Oct 21 2021

Formula

a(n) = a(n-3) + a(n-6) - a(n-9).

a(n) = Sum_{k=0..floor(n/2)} 0^abs(L(C(n-k,2)/3) - 2*L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.

From G. C. Greubel, Nov 18 2021: (Start)

a(n) = A152467(n+3) + A152467(n+6) if n == 1 (mod 3), otherwise A152467(n+6).

a(n) = A175676(n+2) if n == 1 (mod 3), otherwise A152467(n+6).

a(n) = A002264(n+3) if n == 1 (mod 3), otherwise A152467(n+6). (End)