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

Author

Michael Somos, Jan 19 2014

Keywords

• nonn
• easy

Examples

			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 + ...
		

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).

Crossrefs

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

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2-x^2-x^3)/(1-x-x^3-x^4)); // G. C. Greubel, Aug 07 2018

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 *)

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

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