OFFSET
0,3
COMMENTS
a(n) appears in the following formula for the nonnegative 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, called there Q(rho). (rho*sigma)^n = C(n) + B(n)*rho + a(n)*sigma,n>=0, with C(n)= A120757(n) with C(0):=1, and B(n)= |A122600(n-1)| with B(0)=1. For the nonpositive powers see A085810(n)*(-1)^n, A181880(n-2)*(-1)^n and A116423(n+1)*(-1)^(n+1), 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 (3,4,1).
FORMULA
a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), n>=2, a(-1):=1, a(0)=0, a(1)=1.
MATHEMATICA
CoefficientList[Series[x (1+x)/(1-3x-4x^2-x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{3, 4, 1}, {0, 1, 4}, 40] (* Harvey P. Dale, Feb 04 2024 *)
PROG
(PARI) Vec((1+x)/(1-3*x-4*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 26 2010
STATUS
approved