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

1,2

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]".

Links

Table of n, a(n) for n=1..29.

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

Example

(See A192243.)

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 *)

Crossrefs

Cf. A000032, A000045, A192243, A192068.

Partial sums of A081714.

Sequence in context: A098545 A161707 A368773 * A368098 A318395 A151267

Adjacent sequences: A192065 A192066 A192067 * A192069 A192070 A192071

Keyword

nonn

Author

Clark Kimberling, Jun 26 2011

Extensions

New name from Peter Luschny, Mar 10 2015

Status

approved