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%I #21 Feb 04 2024 18:12:11
%S 0,1,4,16,65,263,1065,4312,17459,70690,286218,1158873,4692181,
%T 18998253,76922356,311452261,1261044460,5105864780,20673224441,
%U 83704176903,338911293253,1372223811812,5556020785351,22495868896554,91083913642878,368791237300201,1493205235368669,6045864568949689,24479205885623944,99114281168039257,401305531615563236
%N Expansion of x*(1+x)/(1-3*x-4*x^2-x^3).
%C 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.
%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 (3,4,1).
%F a(n) = 3*a(n-1) + 4*a(n-2) + a(n-3), n>=2, a(-1):=1, a(0)=0, a(1)=1.
%t 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 *)
%o (PARI) Vec((1+x)/(1-3*x-4*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 24 2012
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Nov 26 2010
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