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

Author

Paul Barry, Apr 01 2006

Keywords

• easy
• nonn

Comments

Row sums of A117908.

Links

• 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)

Crossrefs

Cf. A009947, A117908.

Programs

Magma

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

Mathematica

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

Sage

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

Formula

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