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Fibonacci polynomials

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[Triangle of] Fibonacci polynomials
F0 (x)
0
F1 (x)
1
F2 (x)
1
F3 (x)
  1 x + 1
F4 (x)
  2 x + 1
F5 (x)
  1 x 2 +   3 x + 1
F6 (x)
  3 x 2 +   4 x + 1
F7 (x)
1 x 3 +   6 x 2 +   5 x + 1
F8 (x)
4 x 3 + 10 x 2 +   6 x + 1
F9 (x)
1 x 4 + 10 x 3 + 15 x 2 +   7 x + 1
F10 (x)
5 x 4 + 20 x 3 + 21 x 2 +   8 x + 1
F11 (x)
1 x 5 + 15 x 4 + 35 x 3 + 28 x 2 +   9 x + 1
F12 (x)
6 x 5 + 35 x 4 + 56 x 3 + 36 x 2 + 10 x + 1
The sequence of Fibonacci polynomials
Fn (x)
[2] is a sequence of polynomials defined by the recurrence relation

where

  • the degree of 
    Fn (x), n   ≥   1,
    is 
    n / 2⌉  −  1
    ;
  • Fn = Fn (1), n   ≥   0,
    where 
    Fn
    is the 
    n
    th Fibonacci number.


If you look at the Fibonacci polynomials triangle, you will see that the rising diagonals corresponding to odd 
n
are the "(1, 1)-Pascal polynomials". And the column of degree 
1
have natural numbers as coefficients, the column of degree 
2
have triangular numbers as coefficients, the column of degree 
3
have tetrahedral numbers as coefficients, and so on... (Cf. rows of (1, 1)-Pascal triangle, i.e. Pascal triangle.)

This yields the infinite sequence of finite sequences

{{0}, {1}, {1}, {1, 1}, {2, 1}, {1, 3, 1}, {3, 4, 1}, {1, 6, 5, 1}, {4, 10, 6, 1}, {1, 10, 15, 7, 1}, {5, 20, 21, 8, 1}, {1, 15, 35, 28, 9, 1}, {6, 35, 56, 36, 10, 1}, ... },

whose concatenation yields the infinite sequence (see A102426)

{ 0, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 4, 1, 1, 6, 5, 1, 4, 10, 6, 1, 1, 10, 15, 7, 1, 5, 20, 21, 8, 1, 1, 15, 35, 28, 9, 1, 6, 35, 56, 36, 10, 1, ... }.

Fibonacci rabbits per generation

If we interpret the Fibonacci numbers 
Fn
as the number of [male-female] pairs of Fibonacci rabbits at the beginning of the 
n
th month, where each pair that is at least 2 months old begets a [male-female] pair of newborn Fibonacci rabbits, then the coefficient 
ak
of the term 
ak xk
of the Fibonacci polynomial 
Fn (x)
indicates the number of [male-female] pairs of Fibonacci rabbits which belong to the 
k
th generation.

Tree of newborn (red) and mature (green)
"[male-female] pairs of Fibonacci rabbits"
(node labels show the generation
k
)

0 0 0 1 0 1

1 0 1

1

1 2 0 1

1

1 2

1 2

2

For example

means that at the beginning of the 6 th month, we have

  • 3 pairs belonging to the 2 nd generation (offsprings of the offsprings of the original pair),
  • 4 pairs belonging to the 1st generation (offsprings of the original pair),
  • 1 pair belonging to the 0 th generation (the original pair),

for a total of 8 [male-female] pairs of Fibonacci rabbits.

At the beginning of the 
n
th month, we have 
n / 2⌉
generations, i.e. generations 
0
to 
n / 2⌉  −  1
, of [male-female] pairs of Fibonacci rabbits. The original pair, i.e. the 0 th generation, was a [male-female] pair of newborn Fibonacci rabbits, i.e. 0 months old, which was dropped on the island at the beginning of the first month. (At the beginning of the 0 th month, there were no rabbits.)

Reading the rows of generation labels (the labels of row 
n
are the labels of row 
n  −  1
appended with the incremented labels of row 
n  −  2
) left-to-right yields the infinite sequence (see A??????)
{0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 2, 2, ...}.
Reading the rows of generation labels right-to-left yields the infinite sequence (see A200650
(n  +  1), n   ≥   1
)
{0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 2, 1, 1, 1, 0, ...}.

Formulae

where 
T(1, 1) (n, k) = (  nk  )
is the term from column 
k
of row 
n
of the (1, 1)-Pascal triangle (i.e. Pascal triangle), and 
(  nk  )
is a binomial coefficient.

Generating function

The ordinary generating function for the sequence of Fibonacci polynomials is

We may also observe the following relation

where

See also


  • A049310 Triangle of coefficients of Chebyshev polynomials
    S (n, x) := U (n, x / 2)
    (exponents in increasing order). Unsigned triangle 
    | a (n, m) |
    has Fibonacci polynomials (definition according to MathWorld) 
    F (n  +  1, x)
    as row polynomials.



Notes

  1. Weisstein, Eric W., Fibonacci Polynomial, from MathWorld—A Wolfram Web Resource.
  2. MathWorld has a different definition[1] for the sequence of Fibonacci polynomials
    Fn (x)
    (which are related to the Chebyshev polynomials, see A049310), i.e.