A192243 0-sequence of reduction of Lucas sequence by x^2 -> x+1.

1, 1, 5, 12, 34, 88, 233, 609, 1597, 4180, 10946, 28656, 75025, 196417, 514229, 1346268, 3524578, 9227464, 24157817, 63245985, 165580141, 433494436, 1134903170, 2971215072, 7778742049, 20365011073, 53316291173, 139583862444, 365435296162

Offset

1,3

Comments

See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Joerg Arndt, trees described in comment for 1<=n<=5

Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Formula

From Colin Barker, Feb 08 2012: (Start)

G.f.: x*(1-2*x+2*x^2)/(1-3*x+3*x^3-x^4).

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).

(End)

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

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

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

Example

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:
1 -> 1
1+3x -> 1+3x
1+3x+4x^2 -> 1+3x+4(x+1)= 5+7x
1+3x+4x^2+7x^2 -> 12+21x..., so that
0-sequence=(1,1,5,12,...), 1-sequence=(0,3,7,21,...).

Mathematica

c[n_] := LucasL[n]; Table[c[n], {n, 1, 15}]; q[x_] := x + 1; p[0, x_] :=
1; p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1]; reductionRules = {x^y_?EvenQ
-> 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}]
u = Table[Coefficient[Part[t, n], x, 0], {n, 1, 50}] (* A192243 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 50}] (* A192068 *)
(* Peter J. C. Moses, Jun 26 2011 *)
Table[SeriesCoefficient[x*(1 - 2*x + 2*x^2)/(1 - 3*x + 3*x^3 - x^4), {x, 0, n}], {n, 1, 50}]
LinearRecurrence[{3, 0, -3, 1}, {1, 1, 5, 12}, 30] (* G. C. Greubel, Dec 21 2017 *)
Table[If[EvenQ[n], Fibonacci[2*n-1]-1, Fibonacci[2*n-1]], {n, 1, 20}] (* Rigoberto Florez, Aug 29 2019 *)

Prog

(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
(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

Crossrefs

Cf. A192232, A192068.

Sequence in context: A301785 A066280 A333886 * A292104 A136113 A298992

Adjacent sequences: A192240 A192241 A192242 * A192244 A192245 A192246

Keyword

nonn

Author

Clark Kimberling, Jun 26 2011

Status

approved