%I #26 Sep 08 2022 08:45:35

%S 3,1,1,5,7,13,25,45,83,153,281,517,951,1749,3217,5917,10883,20017,

%T 36817,67717,124551,229085,421353,774989,1425427,2621769,4822185,

%U 8869381,16313335,30004901,55187617,101505853,186698371,343391841

%N Expansion of (3-2*x-3*x^2)/(1-x-x^2-x^3).

%H Vincenzo Librandi, <a href="/A141523/b141523.txt">Table of n, a(n) for n = 0..1000</a>

%H Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

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

%F a(0)=3; a(1)=1; a(2)=1; thereafter a(n) = a(n-1) + a(n-2) + a(n-3).

%F From _R. J. Mathar_, Aug 22 2008: (Start)

%F O.g.f.: (3-2*x-3*x^2)/(1-x-x^2-x^3).

%F a(n) = A001644(n) - 2*A000073(n). (End)

%t a[0]=3; a[1]=1; a[2]=1; a[n_]:= a[n]=a[n-1]+a[n-2]+a[n-3]; Table[a[n], {n, 0, 40}]

%t LinearRecurrence[{1, 1, 1}, {3, 1, 1}, 40] (* _Vincenzo Librandi_, Oct 17 2012 *)

%o (Magma) I:=[3, 1, 1]; [n le 3 select I[n] else Self(n-1)+Self(n-2) +Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Oct 17 2012

%o (PARI) a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[3;1;1])[1,1] \\ _Charles R Greathouse IV_, Mar 22 2016

%o (PARI) my(x='x+O('x^40)); Vec((3-2*x-3*x^2)/(1-x-x^2-x^3)) \\ _G. C. Greubel_, Apr 22 2019

%o (Sage) ((3-2*x-3*x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 22 2019

%Y Cf. A000073, A001644.

%K nonn,easy

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 11 2008

%E Edited by _N. J. A. Sloane_, Oct 17 2012