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%I #45 Jul 08 2020 20:37:52
%S 1,-3,13,-52,211,-854,3458,-14001,56689,-229529,929344,-3762837,
%T 15235416,-61686940,249765321,-1011279139,4094585641,-16578638800,
%U 67125538103,-271785755150,1100438056662,-4455582728689,18040286167865,-73043627475013,295747609825188,-1197457625543481
%N Expansion of 1/(1 + 3*x - 4*x^2 + x^3).
%C Suggested by the Steinbach heptagon polynomial p^3 - 2*p^2*(1 - p) - p(1 - p)^2 + (1 - p)^3 = (1 - 4 p + 3 p^2 + p^3).
%C B(n):=|a(n-1)| = a(n-1)*(-1)^(n-1) with B(0):=0 (hence the o.g.f. for B(n) is x/(1 + 3*x - 4*x^2 + x^3)) 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. (rho*sigma)^n = C(n) + B(n)*rho + A(n)*sigma,n>=0, with C(n)= A120757(n) with C(0):=1, and A(n)= A181879(n). For the nonpositive powers see A085810*(-1)^n, A181880(n) and A116423(n)*(-1)^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 (-3,4,-1).
%F a(n)= -3*a(n-1) + 4*a(n-2) - a(n-3), n>=2, a(-1):=0, a(1)=0, a(1)=-3 (from the o.g.f. given in the name).
%F a(n) = (-1)^n*Sum_{k=0..n} binomial(n+k+2,3*k+2)*7^k. - _Emanuele Munarini_, Aug 27 2017
%F From _Kai Wang_, Jul 05 2020: (Start)
%F a(n) = Sum_{i+2j+3k=n} (-1)^(i+k)*3^i*4^j*((i+j+k)!)/(i!*j!*k!).
%F a(n) = (-1)^n*(6*A215076(n+4) - 21*A215076(n+3) - 13*A215076(n+2))/7. (End)
%t p[x_] := 1 - 4 x + 3x^2 + x^3; q[x_] := ExpandAll[x^3*p[1/x]]; Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 30}], n], {n, 0, 30}]
%t CoefficientList[Series[1/(1 + 3*x - 4*x^2 + x^3), {x, 0, 50}], x] (* or *) LinearRecurrence[{-3, 4, -1}, {1, -3, 13}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2012 *)
%Y Cf. A065941.
%K sign,easy
%O 0,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 20 2006
%E Edited by _N. J. A. Sloane_, Feb 01 2007
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