A279277 Composition of Lucas numbers A000032 with Fibonacci numbers A000045.

1, 4, 12, 37, 110, 327, 968, 2864, 8469, 25040, 74029, 218856, 647008, 1912753, 5654670, 16716883, 49420052, 146100276, 431915561, 1276869920, 3774804441, 11159436284, 32990587972, 97529916957, 288327225550, 852380393407, 2519888066928, 7449533000584, 22023018662909

Offset: 1

Author

Oboifeng Dira, Dec 10 2016

Keywords

• nonn
• easy

Comments

G(F(x)) where F(x) = x+x^2+2x^3+3x^4+... is the generating series of the Fibonacci numbers A000045 and G(x) = x+3x^2+4x^3+7x^4 +... is the generating series of the Lucas numbers A000032.

Examples

			(x+x^2)/(1-3x) = x + (3+1)x^2+... so a(1) = 1 and a(2) = 4.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Crossrefs

Cf. A000045, A000032, A270863.

Programs

Mathematica

Rest@ CoefficientList[Series[(x + x^2 - x^3)/(1 - 3 x - x^2 + 3 x^3 + x^4), {x, 0, 24}], x] (* Michael De Vlieger, Dec 12 2016 *)

Formula

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

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4), a(1)=1, a(2)=4, a(3)=12, a(4)=46.