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%I #44 Mar 10 2020 15:03:18
%S 0,1,1,3,6,14,31,70,157,353,793,1782,4004,8997,20216,45425,102069,
%T 229347,515338,1157954,2601899,5846414,13136773,29518061,66326481,
%U 149034250,334876920,752461609,1690765888,3799116465,8536537209
%N Expansion of x*(1-x)/(1-2*x-x^2+x^3).
%C Essentially a duplicate of A077998: a(n) = A077998(n-1). - _Joerg Arndt_, Aug 14 2015
%C a(n) appears in the formula for the nonnegative powers of sigma, the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with rho = 2*cos(Pi/7), the ratio of the smaller heptagon diagonal to the side length, as follows. sigma^n = a(n-1)*1 + B(n)*rho + a(n)*sigma, n>=0, with B(n)=A006054(n). Put a(-1):= 1. See the Steinbach reference, and a comment under A052547.
%C a(n-1) is the top left entry of the n-th power of the 3X3 matrix [0, 1, 0; 1, 1, 1; 0, 1, 1] or of the 3X3 matrix [0, 0, 1; 0, 1, 1; 1, 1, 1]. - _R. J. Mathar_, Feb 03 2014
%H Michael De Vlieger, <a href="/A106803/b106803.txt">Table of n, a(n) for n = 0..2845</a>
%H P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), p. 22-31.
%H Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1).
%F a(n) = A077998(n-1). - _R. J. Mathar_, Aug 07 2008
%F a(n) = A187070(2*n), a(n) = A187068(2*n+2). - _L. Edson Jeffery_, Mar 10 2011
%F a(n+1) = - A199853(n+1). - _G. C. Greubel_, Aug 14 2015
%F a(n) = 2*a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=a(2)=1. - _G. C. Greubel_, Aug 14 2015
%F a(n) = A006356(n-2) for n > 1. - _Georg Fischer_, Oct 21 2018
%t m = {{0, 0, 1}, {1, 2, 0}, {1, 1, 0}}; v[0] = {0, 1, 1}; v[n_] := m.v[n - 1]; Table[v[n][[1]], {n, 0, 30}] (* Edited and corrected by _L. Edson Jeffery_, Oct 18 2017 *)
%t RecurrenceTable[{a[1]== 0, a[2]== 1, a[3]== 1, a[n]== 2*a[n-1] + a[n-2] - a[n-3]}, a, {n,30}] (* _G. C. Greubel_, Aug 14 2015 *)
%o (PARI) concat(0,Vec((1-x)/(x^3-2*x-x^2+1)+O(x^99))) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y Cf. A006356, A077998, A187068, A187070, A199853.
%K nonn,easy,less
%O 0,4
%A _Roger L. Bagula_, May 17 2005
%E Edited by _N. J. A. Sloane_, Aug 08 2008
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