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A123020 Expansion of (1 -5*x +5*x^2)/((1 -2*x)*(1 -4*x +x^2)).

%I A123020 #30 Feb 16 2025 08:33:02
%S A123020 1,1,2,5,14,43,142,493,1766,6443,23750,88045,327406,1219531,4546622,
%T A123020 16958765,63272054,236096683,881049142,3287968813,12270563966,
%U A123020 45793762763,170903438510,637817894125,2380363943686,8883629492011
%N A123020 Expansion of (1 -5*x +5*x^2)/((1 -2*x)*(1 -4*x +x^2)).
%C A123020 Denominator of reduced g.f. is essentially the characteristic polynomial of [1, 1, 0; 1, 2, 1; 0, 1, 3]. - _Paul Barry_, Dec 17 2009
%H A123020 G. C. Greubel, <a href="/A123020/b123020.txt">Table of n, a(n) for n = 0..1000</a>
%H A123020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%H A123020 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,2).
%F A123020 From _Paul Barry_, Dec 17 2009: (Start)
%F A123020 G.f.: 1/(1 -x -x^2/(1 -2*x -x^2/(1-3*x))) = (1-5*x+5*x^2)/(1-6*x+9*x^2-2*x^3).
%F A123020 a(n) = ((2+sqrt(3))/6)*(2-sqrt(3))^n + ((2-sqrt(3))/6)*(2+sqrt(3))^n + 2^n/3. (End)
%F A123020 a(n) = (1/3)*(2^n - ChebyshevT(n+1, 2) + 4*ChebyshevT(n, 2)). - _G. C. Greubel_, Jul 11 2021
%F A123020 3*a(n) = 2^n +A001075(n-1), n>=1. - _R. J. Mathar_, Aug 05 2021
%t A123020 Table[(2^n - ChebyshevT[n + 1, 2] + 4*ChebyshevT[n, 2])/3, {n,0,30}] (* _G. C. Greubel_, Jul 11 2021 *)
%o A123020 (Magma) I:=[1,1,2]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) +2*Self(n-3): n in [1..31]]; // _G. C. Greubel_, Jul 11 2021
%o A123020 (Sage)
%o A123020 def a(n): return (1/3)*(2^n - chebyshev_T(n+1, 2) + 4*chebyshev_T(n, 2))
%o A123020 [a(n) for n in (0..30)] # _G. C. Greubel_, Jul 11 2021
%Y A123020 Cf. A001519, A080937.
%K A123020 nonn,easy
%O A123020 0,3
%A A123020 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 24 2006
%E A123020 Edited by _N. J. A. Sloane_, Jun 13 2007
%E A123020 New name and change of offset by _G. C. Greubel_, Jul 11 2021