A006493 Generalized Lucas numbers.

1, 0, 6, 7, 28, 54, 135, 286, 627, 1313, 2730, 5565, 11212, 22304, 43911, 85614, 165490, 317373, 604296, 1143054, 2149074, 4017950, 7473180, 13832910, 25490115, 46774448, 85494900, 155693873, 282551856, 511101624, 921676437, 1657238030, 2971622493, 5314551351

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Programs

Maple

A006493:=(1-2*z+2*z**2)*(z-1)**3/(z**2+z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
a:= n-> (Matrix([[7,6,0,1,0$4,-2,18]]). Matrix(10, (i,j)-> if (i=j-1) then 1 elif j=1 then [5,-5,-10,15,11, -15,-10,5,5,1][i] else 0 fi)^n)[1,7]: seq (a(n), n=3..36); # Alois P. Heinz, Aug 26 2008

Mathematica

CoefficientList[(1-x)^3*(1-2*x+2*x^2)/(1-x-x^2)^5 + O[x]^40, x] (* Jean-François Alcover, May 29 2015 *)

Formula

G.f. has denominator (1 - x - x^2)^5.