A247035 Expansion of 2*(x+1)*(x^4+6*x^3+5*x^2+6*x+1)/(x^6-18*x^3+1).

2, 14, 22, 58, 266, 398, 1042, 4774, 7142, 18698, 85666, 128158, 335522, 1537214, 2299702, 6020698, 27584186, 41266478, 108037042, 494978134, 740496902, 1938646058, 8882022226, 13287677758, 34787592002, 159381421934, 238437702742, 624238009978

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Mathematics Stack Exchange question, Need formula for sequence related to Lucas/Fibonacci numbers (with answer by Robert Israel).

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Formula

G.f.: 2*(x+1)*(x^4+6*x^3+5*x^2+6*x+1)/(x^6-18*x^3+1).

a(n) = (7/2)*( 3*F(2n)+F(2n-1) ) if n==1 (mod 3); otherwise a(n) = 2*( 3*F(2n)+F(2n-1) ), where F = A000045. [Robert Israel, see Link section]

Mathematica

CoefficientList[Series[2 (x + 1) (x^4 + 6 x^3 + 5 x^2 + 6 x + 1)/(x^6 - 18 x^3 + 1), {x, 0, 30}], x]
LinearRecurrence[{0, 0, 18, 0, 0, -1}, {2, 14, 22, 58, 266, 398}, 30] (* Harvey P. Dale, Jul 27 2018 *)

Prog

(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients (R!(2*x*(x+1)*(x^4+6*x^3+5*x^2+6*x+1)/(x^6-18*x^3+1)));
(Magma) A002878:=func<i | 3*Fibonacci(2*i)+Fibonacci(2*i-1)>; [IsOne(n mod 3) select (7/2)*A002878(n) else 2*A002878(n): n in [0..30]]; // Bruno Berselli, Sep 10 2014

Crossrefs

Cf. A000045, A002878.

Sequence in context: A074312 A061426 A190045 * A069512 A328217 A116639

Adjacent sequences: A247032 A247033 A247034 * A247036 A247037 A247038

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 10 2014

Status

approved