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A073446 Product L(n)*S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) * A001644(n).

%I A073446 #19 Jan 05 2025 19:51:37
%S A073446 6,1,9,28,77,231,702,2059,6157,18316,54489,162185,482678,1436397,
%T A073446 4274853,12722028,37861085,112675763,335326230,997940307,2969899037,
%U A073446 8838503884,26303639349,78280380217,232964641030,693309407681
%N A073446 Product L(n)*S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) * A001644(n).
%C A073446 a(n) is also the trace of the matrix R^n, where R is the Kronecker product of the Fibonacci matrix (Fibomatrix): first row (1,1), second row (1,0), times the Tribomatrix: first row (1,1,0), second row (1,0,1), third row (1,0,0).
%C A073446 a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104576 (an index of a prime Lucas 3-step number). The only known such are n = 2, 4, 7, 8 (through 67661). - _Jonathan Vos Post_, May 10 2005
%D A073446 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%H A073446 Vincenzo Librandi, <a href="/A073446/b073446.txt">Table of n, a(n) for n = 0..1000</a>
%H A073446 M. Elia, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-2/elia.pdf">Derived Sequences, The Tribonacci Recurrence and Cubic Forms</a>, The Fibonacci Quarterly 39.2 (2001): 107-109.
%H A073446 F. T. Howard, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-4/howard1.pdf">A Tribonacci Identity</a>, The Fibonacci Quarterly 39.4 (2001): 352-357.
%H A073446 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,2,-1,1).
%F A073446 a(n) = a(n-1)+4*a(n-2)+5*a(n-3)+2*a(n-4)-a(n-5)+a(n-6), a(0)=6, a(1)=1, a(2)=9, a(3)=28, a(4)=77, a(5)=231.
%F A073446 G.f.: (6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6).
%t A073446 CoefficientList[Series[(6-5x-16x^2-15x^3-4x^4+x^5)/(1-x-4x^2-5x^3-2x^4 +x^5-x^6), {x, 0, 50}], x]
%o A073446 (PARI) my(x='x+O('x^40)); Vec((6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)) \\ _G. C. Greubel_, Feb 19 2019
%o A073446 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6) )); // _G. C. Greubel_, Feb 19 2019
%o A073446 (Sage) ((6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 19 2019
%o A073446 (GAP) a:=[6,1,9,28,77,231];; for n in [7..40] do a[n]:=a[n-1]+4*a[n-2] +5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Feb 19 2019
%Y A073446 Cf. A000032, A000040, A001358, A001606, A001644, A104576.
%K A073446 easy,nonn
%O A073446 0,1
%A A073446 Mario Catalani (mario.catalani(AT)unito.it), Aug 01 2002