A115605 Expansion of -x^2*(2 + x - 2*x^2 - x^3 + 2*x^4) / ( (x-1)*(1+x)*(1 + x + x^2)*(x^2 - x + 1)*(x^2 + 4*x - 1)*(x^2 - x - 1) ).

0, 0, 2, 7, 31, 128, 549, 2315, 9826, 41594, 176242, 746496, 3162334, 13395658, 56745250, 240376201, 1018250793, 4313378176, 18271765435, 77400436781, 327873517634, 1388894499108, 5883451527348, 24922700587008

Offset: 0

Author

Roger L. Bagula, Mar 13 2006

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1,0,1,-3,-6,3,1).

Crossrefs

Cf. A000045, A079962.

Programs

Maple

A000035 := proc(n)
        n mod 2 ;
end proc:
A061347 := proc(n)
        op((n mod 3)+1,[-2,1,1]) ;
end proc:
A001076 := proc(n)
        option remember;
        if n <=1 then
                n;
        else
                4*procname(n-1)+procname(n-2) ;
        end if;
end proc:
A039834 := proc(n)
        (-1)^(n+1)*combinat[fibonacci](n) ;
end proc:
A087204 := proc(n)
        op((n mod 6)+1,[2,1,-1,-2,-1,1]) ;
end proc:
A115605 := proc(n)
        -A000035(n+1)/6 +A061347(n+2)/12 + A001076(n+1)/10 +3*A039834(n+1)/20 -A087204(n)/12 ;
end proc: # R. J. Mathar, Dec 16 2011

Mathematica

LinearRecurrence[{3,6,-3,-1,0,1,-3,-6,3,1},{0,0,2,7,31,128,549,2315,9826,41594},30] (* Harvey P. Dale, Dec 16 2011 *)

PARI

concat([0,0],Vec((2+x-2*x^2-x^3+2*x^4)/((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)*(x^2+4*x-1)*(x^2-x-1))+O(x^99))) \\ Charles R Greathouse IV, Sep 27 2012

Formula

Lim_{n->infinity} a(n+1)/a(n) = phi^3 = A098317.

a(n) = -A000035(n+1)/6 +A061347(n+2)/12 +A001076(n+1)/10 +3*A039834(n+1)/20 -A087204(n)/12. - R. J. Mathar, Dec 16 2011