A181880 Expansion of 1/(1-4*x-3*x^2-x^3).

1, 4, 19, 89, 417, 1954, 9156, 42903, 201034, 942001, 4414009, 20683073, 96916320, 454128508, 2127946065, 9971086104, 46722311119, 218930448853, 1025859814873, 4806952917170, 22524321562152, 105544004814991, 494555936863590, 2317380083461485, 10858732149251701, 50881624784254849, 238420075668235984, 1117183909174960184, 5234877488488803537, 24529481757148330684

Offset: 0

Author

Wolfdieter Lang, Nov 27 2010

Keywords

• nonn
• easy

Comments

B(n):=a(n-2)*(-1)^n, B(0):=0, B(1):=0, (o.g.f. x^2/(1 + 4*x + 3*x^2 -x^3))appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with C(n)= A085810(n)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.

Links

• P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
• Index entries for linear recurrences with constant coefficients, signature (4, 3, 1).

Programs

Mathematica

CoefficientList[Series[1/(1-4*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,1},{1,4,19},40] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

Formula

O.g.f.: 1/(1-4*x-3*x^2-x^3).

a(n) = 4*a(n) + 3*a(n-2) +a(n-3), n>=2, a(-1):=0, a(0)=1, a(1)=4.