| Comments from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2007 (Start): Sequences which equal the sequence of their d-th differences obey linear recurrences with constant binomial coefficients of the form sum_{i=0..d} binomial(d,d-i)*(-1)^i*a(n-i)=a(n-d).
If d is even, this simplifies to sum_{i=0..d-1} binomial(d,d-i)*(-1)^i*a(n-i)=0.
This binding of d (d odd) or d-1 (d even) consecutive terms by the recurrences leaves d or d-1, respectively, free parameters to choose a(0),a(1),...a(d) or a(0),a(1),...a(d-1), respectively, which ultimately define the individual sequence.
The generating functions are
d=2: a(0)/(1-2*x).
d=3: 1/3*(-a(0)+a(1)-a(2))/(-1+2*x)+1/3*(-4*a(0)*x-x*a(2)+4*a(1)*x-a(2)+2*a(0)+a(1))/(x^2-x+1).
d=4: 1/2*(-2*a(0)+2*a(1)-a(2))/(-1+2*x)+1/2*(2*a(1)*x-4*a(0)*x-a(2)+2*a(1))/(1-2*x+2*x^2) .
In the present sequence we have d=3 and g.f. = (x-1)/(x^2-x+1)-2/(-1+2*x) . (End)
Also binomial transform of A130784. a(n)=2^(n+1) + A010892(n+4).
Recurrence in shorter form: a(n)=2a(n) + periodicly extended 2 1 -1 -2 -1 1.
See A130750, A130752, A130755 for other examples of d=3 sequences, A130781 for an example of d=4.
| 