cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

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

Keywords

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.

Crossrefs

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).