A214825 - OEIS

%I #56 Dec 17 2023 16:41:09

%S 1,3,3,7,13,23,43,79,145,267,491,903,1661,3055,5619,10335,19009,34963,

%T 64307,118279,217549,400135,735963,1353647,2489745,4579355,8422747,

%U 15491847,28493949,52408543,96394339,177296831,326099713,599790883,1103187427

%N a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 3.

%C Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index. See Comments in A214727.

%H Indranil Ghosh, <a href="/A214825/b214825.txt">Table of n, a(n) for n = 0..3770</a>

%H Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.pdf">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 G.f.: (1+2*x-x^2)/(1-x-x^2-x^3).

%F a(n) = K(n) - 2*T(n+1) + 4*T(n), where K(n) = A001644(n), and T(n) = A000073(n+1). - _G. C. Greubel_, Apr 23 2019

%t LinearRecurrence[{1,1,1},{1,3,3},40] (* _Harvey P. Dale_, Oct 05 2013 *)

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

%o (PARI) my(x='x+O('x^40)); Vec((1+2*x-x^2)/(1-x-x^2-x^3)) \\ _G. C. Greubel_, Apr 23 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+2*x-x^2)/(1-x-x^2-x^3) )); // _G. C. Greubel_, Apr 23 2019

%o (SageMath) ((1+2*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 23 2019

%o (GAP) a:=[1,3,3];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Apr 23 2019

%Y Cf. A000073, A000213, A000288, A000322, A000383, A001644, A060455, A136175, A141036, A141523, A214825-A214831.

%K nonn,easy

%O 0,2

%A _Abel Amene_, Jul 28 2012