OFFSET
0,2
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).
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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 27 2010
STATUS
approved