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%I #46 Jan 29 2023 06:02:53
%S 1,3,10,35,126,462,1715,6419,24157,91238,345401,1309574,4970070,
%T 18874261,71705865,272491891,1035680954,3936821259,14965658694,
%U 56893879910,216295686467,822315097387,3126323230541,11885921055638
%N Expansion of x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3).
%H G. C. Greubel, <a href="/A122068/b122068.txt">Table of n, a(n) for n = 1..1000</a>
%H Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
%H R. Witula, P. Lorenc, M. Rozanski, and M. Szweda, <a href="https://bibliotekanauki.pl/articles/87290">Sums of the rational powers of roots of cubic polynomials</a>, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,7).
%F From _Roman Witula_, May 16 2014: (Start)
%F a(n) = (1/2)*Sum_{k=0..2}(1 - 1/sqrt(7)*cot(2^k * alpha))* (2*sin(2^k * alpha))^(2n), where alpha := 2*Pi/7.
%F a(n) = (A215007(n) + A215008(n+1) - 2*A215008(n))/2. (End)
%F a(n) = binomial(2*n-1, n-1) + Sum_{k=1..n} (-1)^k*binomial(2*n, n+7*k). - _Greg Dresden_, Jan 28 2023
%p seq(coeff(series(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n =1..30); # _G. C. Greubel_, Oct 03 2019
%t M = {{2,1,0,0,0,0}, {1,2,1,0,0,0}, {0,1,2,1,0,0}, {0,0,1,2,1,0}, {0,0,0, 1,2,1}, {0,0,0,0,1,2}}; v[1] = {1,1,1,1,1,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n,30}]
%t Rest@CoefficientList[Series[x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), {x, 0, 30}], x] (* _G. C. Greubel_, Oct 03 2019 *)
%t LinearRecurrence[{7,-14,7},{1,3,10},30] (* _Harvey P. Dale_, Mar 08 2020 *)
%o (PARI) Vec(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)+O(x^30)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o (Magma) I:=[1,3,10]; [n le 3 select I[n] else 7*(Self(n-1) -2*Self(n-2) + Self(n-3)): n in [1..30]]; // _G. C. Greubel_, Oct 03 2019
%o (Sage)
%o def A122068_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)).list()
%o a=A122068_list(30); a[1:] # _G. C. Greubel_, Oct 03 2019
%o (GAP) a:=[1,3,10];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # _G. C. Greubel_, Oct 03 2019
%Y Cf. A087946, A081567.
%Y Cf. A215007, A215008. - _Roman Witula_, May 16 2014
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Oct 15 2006
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