A117908 -id:A117908 - cp's OEIS frontend

A117909 Count, inserting 0 after every even number.

1, 2, 0, 3, 4, 0, 5, 6, 0, 7, 8, 0, 9, 10, 0, 11, 12, 0, 13, 14, 0, 15, 16, 0, 17, 18, 0, 19, 20, 0, 21, 22, 0, 23, 24, 0, 25, 26, 0, 27, 28, 0, 29, 30, 0, 31, 32, 0, 33, 34, 0, 35, 36, 0, 37, 38, 0, 39, 40, 0, 41, 42, 0, 43, 44, 0, 45, 46, 0, 47, 48, 0, 49, 50, 0, 51, 52, 0, 53, 54, 0

Offset: 0

Paul Barry, Apr 01 2006

Row sums of A117908.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1)

Cf. A009947, A117908.

I:=[1,2,0,3,4,0]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..91]]; // G. C. Greubel, Oct 21 2021

Riffle[Range[60],0,3] (* Harvey P. Dale, Sep 12 2013 *)

def a(n): return (2*n+3)/3 if (n%3==0) else 2*(n+2)/3 if (n%3==1) else 0
[a(n) for n in (0..90)] # G. C. Greubel, Oct 21 2021

G.f.: (1 +2*x +x^3)/(1-x^3)^2.
a(n) = Sum_{k=0..n} 0^abs(L(C(n,2)/3) - 2*L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.
a(n) = sin((n+1)*Pi/3)*((8*n+14)*sin((n+1)*Pi/3) - sqrt(3)*cos(n*Pi))/9. - Wesley Ivan Hurt, Sep 24 2017

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

Original entry on oeis.org

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

Paul Barry, Apr 01 2006

Diagonal sums of A117908.

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

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).

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

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

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 *)

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

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)

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A117909 Count, inserting 0 after every even number.

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