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A215500 a(n) = ((sqrt(5) + 3)^n + (-sqrt(5) -1)^n + (-sqrt(5) + 3)^n + (sqrt(5) - 1)^n) / 2^n.

%I A215500 #26 Sep 08 2022 08:46:03
%S A215500 4,2,10,14,54,112,340,814,2254,5702,15250,39404,104004,270922,711490,
%T A215500 1859134,4873054,12748472,33391060,87394454,228841254,599050102,
%U A215500 1568437210,4106054164,10750060804,28143585362,73681573690,192899714414,505019869254,1322156172352
%N A215500 a(n) = ((sqrt(5) + 3)^n + (-sqrt(5) -1)^n + (-sqrt(5) + 3)^n + (sqrt(5) - 1)^n) / 2^n.
%C A215500 Inverse binomial transform is (-1)^n * a(n). - _Michael Somos_, Jun 02 2014
%H A215500 G. C. Greubel, <a href="/A215500/b215500.txt">Table of n, a(n) for n = 0..1000</a>
%H A215500 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,1).
%F A215500 G.f.: 2*(2-x)*(1+x)*(1-2*x)/((1-3*x+x^2)*(1+x-x^2)). - _Colin Barker_, Aug 19 2012
%F A215500 a(n) = 2*a(n-1)+3*a(n-2)-4*a(n-3)+a(n-4). - _Colin Barker_, Aug 20 2012
%F A215500 a(-n) = A124697(n) if n>0. - _Michael Somos_, Jun 02 2014
%e A215500 G.f. = 4 + 2*x + 10*x^2 + 14*x^3 + 54*x^4 + 112*x^5 + 340*x^6 + ...
%p A215500 A215500 := x -> ((sqrt(5)+3)^x+(-sqrt(5)-1)^x+(-sqrt(5)+3)^x+(sqrt(5)-1)^x)/2^x;
%p A215500 seq(simplify(A215500(i)),i=0..29);
%t A215500 a[n_] := ((Sqrt[5] + 3)^n + (-Sqrt[5] - 1)^n + (-Sqrt[5] + 3)^n + (Sqrt[5] - 1)^n)/2^n; Table[a[n] // Simplify, {n, 0, 29}] (* _Jean-François Alcover_, Jul 02 2013 *)
%t A215500 LinearRecurrence[{2,3,-4,1}, {4, 2, 10, 14}, 50] (* _G. C. Greubel_, Apr 23 2018 *)
%o A215500 (Sage)
%o A215500 def A215500(x) :
%o A215500     return ((sqrt(5)+3)^x+(-sqrt(5)-1)^x+(-sqrt(5)+3)^x+(sqrt(5)-1)^x)/2^x
%o A215500 [A215500(i).round() for i in (0..29)]
%o A215500 (PARI) {a(n) = polsym( (1 + (-1)^(n>0)*x - x^2) * (1 - 3*x + x^2), abs(n))[1 + abs(n)]}; /* _Michael Somos_, Jun 02 2014 */
%o A215500 (Magma) I:=[4,2,10,14]; [n le 4 select I[n] else 2*Self(n-1) + 3*Self(n-2) - 4*Self(n-3) + Self(n-4): n in [1..30]]; // _G. C. Greubel_, Apr 23 2018
%Y A215500 Cf. A051927, A215502, A215503.
%Y A215500 Cf. A124697.
%K A215500 nonn,easy
%O A215500 0,1
%A A215500 _Peter Luschny_, Aug 13 2012