A217778 Expansion of (1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)).

1, 3, 10, 34, 117, 407, 1429, 5055, 17986, 64278, 230473, 828391, 2982825, 10754459, 38811802, 140165322, 506449789, 1830590295, 6618524221, 23933966743, 86562282258, 313102489406, 1132598701585, 4097213146599, 14822370816337, 53623952036787

Offset: 0

Author

Philippe Deléham, Mar 24 2013

Keywords

• nonn
• easy

Comments

A diagonal of the square array A217770.

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Crossrefs

Cf. A217770.

Programs

Magma

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)))); // Bruno Berselli, Mar 28 2013

Mathematica

LinearRecurrence[{8, -21, 20, -5}, {1, 3, 10, 34}, 26] (* Bruno Berselli, Mar 28 2013 *)
CoefficientList[Series[(1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)),{x,0,30}],x] (* Harvey P. Dale, Sep 26 2023 *)

Maxima

makelist(expand(((3+sqrt(5))*(5+sqrt(5))^n-(3-sqrt(5))*(5-sqrt(5))^n+(1+sqrt(5))*(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(4*2^n*sqrt(5))), n, 0, 25); /* Bruno Berselli, Mar 28 2013 */

Formula

G.f.: (1-5*x+7*x^2-3*x^3)/(1-8*x+21*x^2-20*x^3+5*x^4).

a(n) = A081567(n) - A094865(n).

a(n) = A217770(n+1,n).

a(n) = 8*a(n-1) -21*a(n-2) +20*a(n-3) -5*a(n-4) for n>3, a(0)=1, a(1)=3, a(2)=10, a(3)=34.

a(n) = ((3+r)*(5+r)^n-(3-r)*(5-r)^n+(1+r)*(3+r)^n-(1-r)*(3-r)^n)/(4*r*2^n), where r=sqrt(5). [Bruno Berselli, Mar 28 2013]