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

0,2

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).

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

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 *)

Prog

(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
(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

Crossrefs

Third (m=2) column of triangle A073370.

Cf. A001045, A073371.

Sequence in context: A060298 A304975 A226546 * A305023 A026573 A326660

Adjacent sequences: A073369 A073370 A073371 * A073373 A073374 A073375

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Status

approved