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A181880 Expansion of 1/(1-4*x-3*x^2-x^3). 5


%S 1,4,19,89,417,1954,9156,42903,201034,942001,4414009,20683073,

%T 96916320,454128508,2127946065,9971086104,46722311119,218930448853,

%U 1025859814873,4806952917170,22524321562152,105544004814991,494555936863590,2317380083461485,10858732149251701,50881624784254849,238420075668235984,1117183909174960184,5234877488488803537,24529481757148330684

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

%C 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.

%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, 3, 1).

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

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

%t 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 *)

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 27 2010

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