A073372 Second convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 3, 12, 34, 99, 261, 678, 1692, 4149, 9959, 23568, 55014, 127031, 290457, 658602, 1482240, 3314025, 7365915, 16285300, 35832810, 78500811, 171293293, 372412782, 806963364, 1743173469, 3754782351, 8066319768, 17285917742, 36957928479, 78847115649

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,3,-11,-6,12,8).

Crossrefs

Third (m=2) column of triangle A073370.

Cf. A001045, A073371.

Programs

Magma

[(2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162: n in [0..40]]; // G. C. Greubel, Sep 28 2022

Mathematica

CoefficientList[Series[-(-1+x+2x^2)^(-3),{x,0,78}],x] (* or *) Table[(-3*(-1)^n*n^2+3*2^(n+2)*n^2-15*(-1)^n*n+9*2^(n+2)*n-16*(-1)^n+2^(n+4))/162,{n,42}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

SageMath

def A073372(n): return (2^(n+3)*(16+15*n+3*n^2) +(-1)^n*(34+21*n+3*n^2))/162
[A073372(n) for n in range(40)] # G. C. Greubel, Sep 28 2022

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073371(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2) * binomial(n-k, k) * 2^k.

a(n) = ((30+9*n)*(n+1)*U(n+1) + 2*(33+9*n)*(n+2)*U(n))/162 with U(n) = A001045(n+1), n>=0.

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

E.g.f.: (1/162)*(32*(4 + 9*x + 3*x^2)*exp(2*x) + (34 - 24*x + 3*x^2)*exp(-x)). - G. C. Greubel, Sep 28 2022