A006493 Generalized Lucas numbers. (Formerly M4063)

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

3,3

References

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

Links

Table of n, a(n) for n=3..36.

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

Formula

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

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 *)

Crossrefs

Sequence in context: A042419 A037956 A095369 * A323133 A348950 A292106

Adjacent sequences: A006490 A006491 A006492 * A006494 A006495 A006496

Keyword

nonn

Author

N. J. A. Sloane

Status

approved