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A078013 Expansion of (1-x)/(1-x+x^3). 5

%I #25 Sep 08 2022 08:45:08

%S 1,0,0,-1,-1,-1,0,1,2,2,1,-1,-3,-4,-3,0,4,7,7,3,-4,-11,-14,-10,1,15,

%T 25,24,9,-16,-40,-49,-33,7,56,89,82,26,-63,-145,-171,-108,37,208,316,

%U 279,71,-245,-524,-595,-350,174,769,1119,945,176,-943,-1888,-2064,-1121,767,2831,3952,3185,354,-3598,-6783,-7137

%N Expansion of (1-x)/(1-x+x^3).

%H G. C. Greubel, <a href="/A078013/b078013.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1).

%F G.f.: (1-x)/(1-x+x^3).

%F a(n) = -A050935(n).

%F a(n+1) = a(n) - a(n-2), a(0)=1, a(1)=a(2)=0. - _Philippe Deléham_, Nov 12 2011

%F G.f.: 1 - x^3 - x^4 + x^6 + x^6/(G(0) - 1) where G(k) = 1 - x*(k+1)/(1 - x/(x - (k+1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 17 2012

%t LinearRecurrence[{1,0,-1}, {1,0,0}, 70] (* or *) CoefficientList[Series[ (1-x)/(1-x+x^3), {x,0,70}], x] (* _G. C. Greubel_, Jun 29 2019 *)

%o (PARI) Vec((1-x)/(1-x+x^3)+O(x^70)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1-x)/(1-x+x^3) )); // _G. C. Greubel_, Jun 29 2019

%o (Sage) ((1-x)/(1-x+x^3)).series(x, 70).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 29 2019

%o (GAP) a:=[1,0,0];; for n in [4..70] do a[n]:=a[n-1]-a[n-3]; od; a; # _G. C. Greubel_, Jun 29 2019

%Y Cf. A050935.

%K sign,easy

%O 0,9

%A _N. J. A. Sloane_, Nov 17 2002

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)