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

6, 1, 9, 28, 77, 231, 702, 2059, 6157, 18316, 54489, 162185, 482678, 1436397, 4274853, 12722028, 37861085, 112675763, 335326230, 997940307, 2969899037, 8838503884, 26303639349, 78280380217, 232964641030, 693309407681

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 01 2002

Keywords

• easy
• nonn

Comments

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).

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

References

• Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Elia, Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly 39.2 (2001): 107-109.
• F. T. Howard, A Tribonacci Identity, The Fibonacci Quarterly 39.4 (2001): 352-357.
• Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).

Crossrefs

Cf. A000032, A000040, A001358, A001606, A001644, A104576.

Programs

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

Magma

m:=40; R:=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

Mathematica

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]

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

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

Formula

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.

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).