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A174192 Expansion of (1-x+2x^2)/ ((1-x)*(1-2x-x^2)). 1

%I

%S 1,2,7,18,45,110,267,646,1561,3770,9103,21978,53061,128102,309267,

%T 746638,1802545,4351730,10506007,25363746,61233501,147830750,

%U 356895003,861620758,2080136521,5021893802,12123924127

%N Expansion of (1-x+2x^2)/ ((1-x)*(1-2x-x^2)).

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1)

%F a(n)=2*a(n-1)+a(n-2)+2, with a(0)=1, a(1)=2.

%F a(n)=-1+[1+sqrt(2)]^n+[1-sqrt(2)]^n+(1/4)*sqrt(2)*{[1+sqrt(2)]^n-[1-sqrt(2)]^n}, with n>=0 [From _Paolo P. Lava_, Mar 17 2010]

%e a(2) = 2*a(1)+a(0)+2 = 2*2+1+2 = 7

%e a(3) = 2*a(2)+a(1)+2 = 2*7+2+2 = 18.

%t CoefficientList[Series[(1-x+2x^2)/((1-x)*(1-2x-x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{3,-1,-1},{1,2,7},30] (* _Harvey P. Dale_, Jul 18 2019 *)

%Y Cf. A174191, A048655 (first differences)

%K nonn

%O 0,2

%A _Clark Kimberling_, Mar 11 2010

%E Terms corrected by R. J. Mathar, Oct 26 2011

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Last modified November 22 13:47 EST 2019. Contains 329393 sequences. (Running on oeis4.)