A192068 a(n) = Fibonacci(2*n) - (n mod 2).

0, 3, 7, 21, 54, 144, 376, 987, 2583, 6765, 17710, 46368, 121392, 317811, 832039, 2178309, 5702886, 14930352, 39088168, 102334155, 267914295, 701408733, 1836311902, 4807526976, 12586269024, 32951280099, 86267571271, 225851433717, 591286729878

Offset: 1

Author

Clark Kimberling, Jun 26 2011

Keywords

• nonn

Comments

Previous name was: 1-sequence of reduction of Lucas sequence by x^2 -> x+1.

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

Examples

			(See A192243.)
		

Crossrefs

Cf. A000032, A000045, A192243, A192068.

Partial sums of A081714.

Programs

Maple

a := n -> combinat[fibonacci](2*n)-(n mod 2):
seq(a(n), n=1..29); # Peter Luschny, Mar 10 2015

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,
   30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192243 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192068 *)
(* Peter J. C. Moses, Jun 26 2011 *)
Table[Fibonacci[2n]-Mod[n,2],{n,30}] (* Harvey P. Dale, Jul 11 2020 *)

Formula

Empirical g.f. and recurrence: x^2*(3-2*x)/(1-3*x+3*x^3-x^4), a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - Colin Barker, Feb 08 2012

a(n) = Fibonacci(2*n) - (n mod 2). - Peter Luschny, Mar 10 2015

Extensions

New name from Peter Luschny, Mar 10 2015