A214727 - OEIS

%I #74 Nov 21 2024 18:03:45

%S 1,2,2,5,9,16,30,55,101,186,342,629,1157,2128,3914,7199,13241,24354,

%T 44794,82389,151537,278720,512646,942903,1734269,3189818,5866990,

%U 10791077,19847885,36505952,67144914,123498751,227149617,417793282

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

%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.

%C Note: A000073 (with offset=1), 1 followed by A000073, A000213, A141523, A214727, A214825 to A214831 completely define possible sequences with a(0)=0,1,2...9 and a(1)=a(2)=0,1,2...9 excluding any multiples of these sequences and the trivial case of a(0)=a(1)=a(2)=0.

%C Note: allowing a(0)=0 and a(1)=a(2)=1,2,3....9 leads to A000073 (with offset=1) and its multiples.

%C Note: allowing a(0)=1,2,3....9 a(1)=a(2)=0 leads to 1 followed by A000073 and its multiples.

%C With offset of 6 this sequence is the 8th row of tribonacci array A136175.

%H Reinhard Zumkeller, <a href="/A214727/b214727.txt">Table of n, a(n) for n = 0..1000</a>

%H Martin Burtscher, Igor Szczyrba, and 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 G.f.: (1+x-x^2)/(1-x-x^2-x^3).

%F a(n) = K(n) -2*T(n+1) + 3*T(n), where K(n) = A001644(n), T(n) = A000073(n+1). - _G. C. Greubel_, Apr 23 2019

%F a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-r^2+2*r+1). - _Fabian Pereyra_, Nov 20 2024

%e G.f. = 1 + 2*x + 2*x^2 + 5 x^3 + 9*x^4 + 16*x^5 + 30*x^6 + 55*x^7 + ...

%t LinearRecurrence[{1,1,1},{1,2,2},40] (* _Ray Chandler_, Dec 08 2013 *)

%o (Haskell)

%o a214727 n = a214727_list !! n

%o a214727_list = 1 : 2 : 2 : zipWith3 (\x y z -> x + y + z)

%o    a214727_list (tail a214727_list) (drop 2 a214727_list)

%o -- _Reinhard Zumkeller_, Jul 31 2012

%o (PARI) a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;2;2])[1,1] \\ _Charles R Greathouse IV_, Mar 22 2016

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

%o (SageMath) ((1+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:=[1,2,2];; 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. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

%K nonn,easy

%O 0,2

%A _Abel Amene_, Jul 27 2012