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%I #19 Sep 08 2022 08:46:17
%S 3,2,5,10,17,32,59,108,199,366,673,1238,2277,4188,7703,14168,26059,
%T 47930,88157,162146,298233,548536,1008915,1855684,3413135,6277734,
%U 11546553,21237422,39061709,71845684,132144815,243052208,447042707,822239730,1512334645,2781617082,5116191457,9410143184
%N a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 3, a(1) = 2, a(2) = 5.
%C The tetranacci sequence(s) (T(n+3) = T(n+2) + T(n+1) + T(n)) have characteristic roots such that one is real and the other two have imaginary components. The real root is given by r = (1/3)*(1 + x + y), where x = (19 + 3*sqrt(33))^(1/3) and y = (19 - 3*sqrt(33))^(1/3). By increasing the power of this root (r^n) it is found that r^n = (1/3)*( A001644(n) + (1/3)*a(n-1)*(x + y) + (1/3)*A000073(n-2)*(x^2 + y^2)). Similar results can be found for the complex roots.
%H G. C. Greubel, <a href="/A274761/b274761.txt">Table of n, a(n) for n = 0..1000</a>
%H Pin-Yen Lin, <a href="http://www.fq.math.ca/Scanned/26-2/lin.pdf">De Moivre type identities for the Tribonacci numbers</a>, The Fibonacci Quarterly 26, no.2, (1988), 131-134
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1).
%F a(n) = a(n-1) + a(n-2) + a(n-3).
%F a(n) = 3*A000073(n) - A000073(n-1).
%F G.f.: (3 - x)/(1 - x - x^2 - x^3).
%e a(4) = a(3) + a(2) + a(1) = 10 + 5 + 2 = 17. - _Michael B. Porter_, Jul 05 2016
%t CoefficientList[Series[(3-x)/(1-x-x^2-x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 1, 1}, {3, 2, 5}, 40]
%o (PARI) Vec((3-x)/(1-x-x^2-x^3) + O(x^40)) \\ _Altug Alkan_, Jul 04 2016
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (3-x)/(1-x-x^2-x^3) )); // _G. C. Greubel_, Apr 23 2019
%o (Sage) ((3-x)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 23 2019
%o (GAP) a:=[3,2,5];; 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, A001644.
%K nonn
%O 0,1
%A _G. C. Greubel_, Jul 04 2016
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