A236144 a(n) = F(floor( (n+3)/2 )) * L(floor( (n+2)/2 )) where F=Fibonacci and L=Lucas numbers.

2, 2, 1, 2, 6, 9, 12, 20, 35, 56, 88, 143, 234, 378, 609, 986, 1598, 2585, 4180, 6764, 10947, 17712, 28656, 46367, 75026, 121394, 196417, 317810, 514230, 832041, 1346268, 2178308, 3524579, 5702888, 9227464, 14930351, 24157818, 39088170, 63245985, 102334154

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Formula

G.f.: (2 - x^2 - x^3) / (1 - x - x^3 - x^4) = (1 - x) * (2 + 2*x + x^2) / ((1 + x^2) * (1 - x - x^2)).

a(n) = a(n-1) + a(n-3) + a(n-4) for all n in Z.

0 = a(n)*a(n+2) + a(n+1)*(+a(n+2) -a(n+3)) for all n in Z.

a(n) = A115008(n+2) - A115008(n+1).

a(n) = A115339(n) * A115339(n-1).

a(2*n - 1) = F(n+1) * L(n-1) = A128535(n+1). a(2*n) = F(n+1) * L(n) = A128534(n+1).

a(n) = A000045(n+1)+A057077(n). - R. J. Mathar, Sep 24 2021

Example

G.f. = 2 + 2*x + x^2 + 2*x^3 + 6*x^4 + 9*x^5 + 12*x^6 + 20*x^7 + 35*x^8 + ...

Mathematica

a[ n_] := Fibonacci[ Quotient[ n + 3, 2]] LucasL[ Quotient[ n, 2]];
CoefficientList[Series[(2-x^2-x^3)/(1-x-x^3-x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 07 2018 *)

Prog

(PARI) {a(n) = fibonacci( (n+3)\2 ) * (fibonacci( n\2+1 ) + fibonacci( n\2-1 ))};
(PARI) x='x+O('x^60); Vec((2-x^2-x^3)/(1-x-x^3-x^4)) \\ G. C. Greubel, Aug 07 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(2-x^2-x^3)/(1-x-x^3-x^4)); // G. C. Greubel, Aug 07 2018

Crossrefs

Cf. A000032, A000045, A115008, A115339, A128534, A128535.

Sequence in context: A368836 A336823 A375062 * A226328 A307599 A162663

Adjacent sequences: A236141 A236142 A236143 * A236145 A236146 A236147

Keyword

nonn,easy

Author

Michael Somos, Jan 19 2014

Status

approved