A164267 A Fibonacci convolution.

0, 1, 2, 7, 16, 46, 114, 309, 792, 2101, 5456, 14356, 37468, 98281, 256998, 673323, 1761984, 4614226, 12078110, 31624285, 82787980, 216750601, 567446112, 1485616392, 3889356696, 10182528721, 26658108074, 69791991919, 182717549872

Offset: 0

Author

Paul Barry, Aug 11 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1)

Programs

Magma

I:=[0,1,2,7]; [n le 4 select I[n] else 2*Self(n-1)+3*Self(n-2)-4*Self(n-3)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 13 2017

Mathematica

LinearRecurrence[{2,3,-4,1},{0,1,2,7},30] (* Harvey P. Dale, Jul 12 2011 *)
CoefficientList[Series[x / ((1 + x - x^2) (1 - 3 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 13 2017 *)

PARI

x='x+O('x^50); concat([0], Vec(x/((1+x-x^2)*(1-3*x+x^2)))) \\ G. C. Greubel, Sep 12 2017

Formula

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

a(n) = Sum_{k=0..n} (-1)^k*F(k+1)*F(2(n-k)).

a(n) = Sum_{k=0..n} C(n,k)*F(k+1)*(1-(-1)^(n-k))/2.

a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) + a(n-4).

a(n) = (A122367(n) - A039834(n-1))/2. - R. J. Mathar, Aug 17 2009