A247565 a(n) = 5*a(n-1) - 10*a(n-2) + 8*a(n-3) with a(0) = 2, a(1) = a(2) = 3.

2, 3, 3, 1, -1, 9, 63, 217, 527, 969, 1311, 1081, 47, -87, 7743, 39961, 121679, 270729, 456543, 548857, 344687, -112791, 380031, 5785561, 24225167, 66310473, 135585183, 208622521, 217744559, 87179049, -72570177, 507315097, 3959709647, 14144835849, 35185603551

Offset: 0

Author

Michael Somos, Sep 20 2014

Keywords

• sign
• easy

Examples

			G.f. = 2 + 3*x + 3*x^2 + x^3 - x^4 + 9*x^5 + 63*x^6 + 217*x^7 + 527*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (5,-10,8).

Crossrefs

Cf. A247560, A247564.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2-7*x+8*x^2)/(1-5*x+10*x^2-8*x^3))); // G. C. Greubel, Aug 04 2018

Mathematica

CoefficientList[Series[(2-7*x+8*x^2)/(1-5*x+10*x^2-8*x^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{5,-10,8}, {2,3,3}, 60] (* G. C. Greubel, Aug 04 2018 *)

PARI

{a(n) = 2^n + real( (1 + quadgen(-7))^n )};

PARI

Vec((2 - 7*x + 8*x^2) / (1 - 5*x + 10*x^2 - 8*x^3) + O(x^50)) \\ Michel Marcus, Sep 22 2014

Formula

G.f.: (2 - 7*x + 8*x^2) / (1 - 5*x + 10*x^2 - 8*x^3).

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

a(n) = 2^n + A247560(n) for all n in Z.

a(n) = A247564(n+1) * A247564(n) for all n in Z.

0 = a(n)*(+4*a(n+1) + 2*a(n+2)) + a(n+1)*(-5*a(n+1) + a(n+2)) for all n in Z.