A262735 Expansion of x*(1-x-x^2)/((1-x)*(1-x-2*x^2-x^3)).

0, 1, 1, 2, 4, 8, 17, 36, 77, 165, 354, 760, 1632, 3505, 7528, 16169, 34729, 74594, 160220, 344136, 739169, 1587660, 3410133, 7324621, 15732546, 33791920, 72581632, 155898017, 334853200, 719230865, 1544835281, 3318150210, 7127051636, 15308187336

Offset: 0

Author

Werner Schulte, Sep 29 2015

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-1).

Crossrefs

Cf. A027907, A077864.

Programs

Maple

a:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif n=3 then 2 else 2*a(n-1)+a(n-2)-a(n-3)-a(n-4); fi; end:  seq(f(n), n=0..50); # Wesley Ivan Hurt, Oct 10 2015

Mathematica

CoefficientList[Series[x (1 - x - x^2)/((1 - x) (1 - x - 2 x^2 - x^3)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 29 2015 *)
LinearRecurrence[{2,1,-1,-1},{0,1,1,2},40] (* Harvey P. Dale, Sep 23 2019 *)

PARI

concat(0, Vec(x*(1-x-x^2)/((1-x)*(1-x-2*x^2-x^3)) + O(x^50))) \\ Michel Marcus, Sep 29 2015

Formula

G.f.: A(x) = x*(1-x-x^2)*B(x), where B is g.f. of A077864.

a(n) = A077864(n+1)-2*A077864(n), n >= 0.

a(n+3) = A077864(n+2)-A077864(n+1)-A077864(n), n >= 0.

Recurrence: a(0)=0, a(1)=1, a(2)=1, a(3)=2, a(4)=4, a(5)=8, a(6)=17, and a(n) = 4*a(n-1)-4*a(n-2)+a(n-7) for n >= 7.

Conjecture: a(n+1) = Sum_{j=0..n/2} A027907(n-j,2*j), n >= 0.

a(n) = 2*a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - Wesley Ivan Hurt, Oct 10 2015

a(n) = a(n-1)+2*a(n-2)+a(n-3)-1, n>=3. - R. J. Mathar, Nov 07 2015