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

%I #28 Sep 08 2022 08:46:09

%S 2,14,22,58,266,398,1042,4774,7142,18698,85666,128158,335522,1537214,

%T 2299702,6020698,27584186,41266478,108037042,494978134,740496902,

%U 1938646058,8882022226,13287677758,34787592002,159381421934,238437702742,624238009978

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

%H Vincenzo Librandi, <a href="/A247035/b247035.txt">Table of n, a(n) for n = 0..1000</a>

%H Mathematics Stack Exchange question, <a href="http://math.stackexchange.com/questions/193973/need-formula-for-sequence-related-to-lucas-fibonacci-numbers">Need formula for sequence related to Lucas/Fibonacci numbers</a> (with answer by _Robert Israel_).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,18,0,0,-1).

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

%F 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]

%t 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]

%t LinearRecurrence[{0,0,18,0,0,-1},{2,14,22,58,266,398},30] (* _Harvey P. Dale_, Jul 27 2018 *)

%o (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)));

%o (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

%Y Cf. A000045, A002878.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Sep 10 2014