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Tribonacci-like sequence; a(0)=2, a(1)=1, a(2)=1, a(n) = a(n-1) + a(n-2) + a(n-3).
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%I #48 Sep 08 2022 08:45:35

%S 2,1,1,4,6,11,21,38,70,129,237,436,802,1475,2713,4990,9178,16881,

%T 31049,57108,105038,193195,355341,653574,1202110,2211025,4066709,

%U 7479844,13757578,25304131,46541553,85603262,157448946,289593761

%N Tribonacci-like sequence; a(0)=2, a(1)=1, a(2)=1, a(n) = a(n-1) + a(n-2) + a(n-3).

%C I used the short MATLAB program from the zip file link altered to produce a Lucas version of the tribonacci numbers.

%C No term is divisible by 8 or 9. - _Vladimir Joseph Stephan Orlovsky_, Mar 24 2011

%C a(A246517(n)) = A246518(n). - _Reinhard Zumkeller_, Sep 15 2014

%D Martin Gardner, Mathematical Circus, Random House, New York, 1981, p. 165.

%H Robert Price, <a href="/A141036/b141036.txt">Table of n, a(n) for n = 0..1000</a>

%H Martin Burtscher, Igor Szczyrba, RafaƂ 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 T.-X. He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/He/he13.html">Impulse Response Sequences and Construction of Number Sequence Identities</a>, J. Int. Seq. 16 (2013) #13.8.2

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

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

%F From _R. J. Mathar_, Aug 04 2008: (Start)

%F a(n) = 2*A000213(n) - A000073(n+1).

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

%t a[0]=2; 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}] (* _Alonso del Arte_, Mar 24 2011 *)

%t LinearRecurrence[{1, 1, 1}, {2, 1, 1}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2011 *)

%o (Haskell)

%o a141036 n = a141036_list !! n

%o a141036_list = 2 : 1 : 1 : zipWith3 (((+) .) . (+))

%o a141036_list (tail a141036_list) (drop 2 a141036_list)

%o -- _Reinhard Zumkeller_, Sep 15 2014

%o (PARI) a(n)=([0,1,0;0,0,1;1,1,1]^n*[2;1;1])[1,1] \\ _Charles R Greathouse IV_, Jun 15 2015

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

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-x-2*x^2)/(1-x-x^2-x^3) )); // _G. C. Greubel_, Apr 22 2019

%o (Sage) ((2-x-2*x^2)/(1-x-x^2-x^3)).series(x, 41).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 22 2019

%Y Cf. A000073, A000213, A001644 (Lucas tribonacci sequence), A246517, A246518.

%K nonn,easy

%O 0,1

%A Matt Wynne (matwyn(AT)verizon.net), Jul 30 2008

%E Corrected offset and indices in formulas, _R. J. Mathar_, Aug 05 2008

%E Better name from _T. D. Noe_, Aug 06 2008