%I #48 Sep 08 2022 08:45:57
%S 1,1,5,12,34,88,233,609,1597,4180,10946,28656,75025,196417,514229,
%T 1346268,3524578,9227464,24157817,63245985,165580141,433494436,
%U 1134903170,2971215072,7778742049,20365011073,53316291173,139583862444,365435296162
%N 0-sequence of reduction of Lucas sequence by x^2 -> x+1.
%C See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%C Number of rooted ordered trees with n non-root nodes such that successive branch heights are weakly decreasing; examples are given in the Arndt link. - _Joerg Arndt_, Aug 27 2014
%H G. C. Greubel, <a href="/A192243/b192243.txt">Table of n, a(n) for n = 1..1000</a>
%H Joerg Arndt, <a href="/A192243/a192243.txt">trees described in comment for 1<=n<=5</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F From _Colin Barker_, Feb 08 2012: (Start)
%F G.f.: x*(1-2*x+2*x^2)/(1-3*x+3*x^3-x^4).
%F a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
%F (End)
%F a(n) = (-1)*(2^(-1-n)*(5*((-2)^n+2^n) + (-5+sqrt(5))*(3+sqrt(5))^n - (3-sqrt(5))^n*(5 + sqrt(5)))) / 5. - _Colin Barker_, Dec 22 2017
%F a(n) = F(2n-1)-1 if n is even and F(2n-1) if n is odd, where F(n) is the n-th Fibonacci number. - _Rigoberto Florez_, Aug 29 2019
%F E.g.f.: - cosh(x) + (1/5)*(cosh(3*x/2) + sinh(3*x/2))*(5*cosh(sqrt(5)*x/2) - sqrt(5)*sinh(sqrt(5)*x/2)). - _Stefano Spezia_, Aug 30 2019
%e The Lucas sequence provides coefficients for the power series 1+3x+4x^2+7x^3+..., whose partial sums are polynomials to which we apply reduction by x^2 -> x+1 as introduced at A192232:
%e 1 -> 1
%e 1+3x -> 1+3x
%e 1+3x+4x^2 -> 1+3x+4(x+1)= 5+7x
%e 1+3x+4x^2+7x^2 -> 12+21x..., so that
%e 0-sequence=(1,1,5,12,...), 1-sequence=(0,3,7,21,...).
%t c[n_] := LucasL[n]; Table[c[n], {n, 1, 15}]; q[x_] := x + 1; p[0, x_] :=
%t 1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]; reductionRules = {x^y_?EvenQ
%t -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 50}]
%t u = Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}] (* A192243 *)
%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 50}] (* A192068 *)
%t (* _Peter J. C. Moses_, Jun 26 2011 *)
%t Table[SeriesCoefficient[x*(1 - 2*x + 2*x^2)/(1 - 3*x + 3*x^3 - x^4), {x, 0, n}], {n, 1, 50}]
%t LinearRecurrence[{3,0,-3,1}, {1,1,5,12}, 30] (* _G. C. Greubel_, Dec 21 2017 *)
%t Table[If[EvenQ[n],Fibonacci[2*n-1]-1, Fibonacci[2*n-1]], {n,1,20}] (* _Rigoberto Florez_, Aug 29 2019 *)
%o (PARI) x='x+O('x^30); Vec(x*(1-2*x+2*x^2)/(1-3*x+3*x^3-x^4)) \\ _G. C. Greubel_, Dec 21 2017
%o (Magma) I:=[1, 1, 5, 12]; [n le 4 select I[n] else 3*Self(n-1) - 3*Self(n-3) + Self(n-4): n in [1..30]]; // _G. C. Greubel_, Dec 21 2017
%Y Cf. A192232, A192068.
%K nonn
%O 1,3
%A _Clark Kimberling_, Jun 26 2011