The Lucas-nacci Numbers

Jelly, 12 bytes

;2U+¥Ð¡-,1ZḢ

Try it online!

Background

We can extend F and L to -1 by defining F(-1) = 1 and L(-1) = -1. This is consistent with the recursive function.

Our program starts with

-1  1
 0  2

In each step, to form the next pair, we reverse the last pair and add it to the penultimate pair. For example:

[0, 2] U+¥ [-1, 1] -> [2, 0] + [-1, 1] -> [1, 1]

If we continue this process for a few more steps, we get

-1  1
 0  2
 1  1
 1  3
 4  2
 3  7
11  5

The Lucas-nacci sequence is simply the left column.

How it works

;2U+¥Ð¡-,1ZḢ  Niladic link. No implicit input.
              Since the link doesn't start with a nilad, the argument 0 is used.

;2            Concatenate the argument with 2, yielding [0, 2].
       -,1    Yield [-1, 1]. This is [L(-1), F(-1)].
    ¥         Create a dyadic chain of the two atoms to the left:
  U             Reverse the left argument.
   +            Add the reversed left argument and the right one, element-wise.
     С       For reasons‡, read a number n from STDIN.
              Repeatedly call the dyadic link U+¥, updating the right argument with
              the value of the left one, and the left one with the return value.
              Collect all intermediate results.
          Z   Zip the list of results, grouping the first and seconds coordinates.
           Ḣ  Head; select the list of first coordinates.

^‡ С peeks at the two links to the left: 2 and U+¥. Since the leftmost one is a nilad, it cannot be the body of the loop. Therefore, U+¥ is used as body and a number is read from input. Since there are no command-line arguments, that number is read from STDIN.

Haskell, 59, 57, 56, 52, 51 bytes

l a=2*mod a 2:scanl(+)1(l a)
f n=[l i!!i|i<-[0..n]]

Series definition adapted from this answer.

Less golfed:

fibLike start = start : scanl (+) 1 (fibLike start)
whichStart i = (2*mod i 2)
lucasNacci i = fibLike (whichStart i) !! i
firstN n = [ lucasNacci i | i <- [0..n]]

fibLike start gives an infinite list, defined: f(0)=start, f(1)=1, f(n)=f(n-1) + f(n-2). whichStart i returns 2 for odd input (Lucas series) or 0 for even (Fibonacci series). lucasNacci i gives the ith Lucas-nacci number. firstN n maps over the list.

One byte saved by Boomerang.

CJam, 21 20 bytes

Thanks to Sp3000 for saving 1 byte.

TXX4ri{1$3*4$-}*?;]p

Test it here.

Explanation

Simply uses the recurrence given in the challenge spec.

TXX4 e# Push 0 1 1 4 as base cases.
ri   e# Read input and convert to integer N.
{    e# Run this N times...
  1$ e#   Copy a(n-2).
  3* e#   Multiply by 3.
  4$ e#   Copy a(n-4).
  -  e#   Subtract.
}*
?;   e# Discard the last three values, using a ternary operation and popping the result.
]p   e# Wrap the rest in an array and pretty-print it.