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Coefficients of 子葉さん's k-Fibonacci formula

n terms of a linear combination of $k$ Lucas numbers $L_{n}^{[k]}$ .

$F_{n}^{[k]} = \frac{k - 1}{2 (2 k)^{k} - (k + 1)^{k + 1}} (\sum_{j = 1}^{k - 1} (k + 1)^{k - 1 - j} ((k + 1)^{j} - 2 (2 k)^{j - 1}) L_{n - j + 1}^{[k]} + \frac{2 ((2 k)^{k - 1} - (k + 1)^{k - 1})}{k - 1} L_{n - k + 1}^{[k]})$

The coefficients of those Lucas numbers seem to appear naturally in the matrix setting .
(子葉さん has provided a proof in the comments based on his nice computations in a previous article )

Let $p (x) = x^{k} - \sum_{i = 0}^{k - 1} x^{i}$ be the characteristic polynomial of the companion matrix
$M = (\begin{matrix} 1 \\ ⋱ \\ 1 \\ 1 & \dots & \dots & 1 \end{matrix})$

If we define the coefficients $c_{0}, . . ., c_{k - 1}$ as follows:
$(\begin{matrix} * \\ c_{k - 1} & \dots & c_{0} \end{matrix}) = p^{'} (M)^{- 1}$
then the following identity holds
$F_{n}^{[k]} = \sum_{i = 0}^{k - 1} c_{i} L_{n - i}^{[k]}$

$k$	$M$	$p^{'} (M)^{- 1}$	子葉's $F_{n}^{[k]}$ formula
2	$(\begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array})$	$(\begin{array}{rr} - \frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array})$	$\frac{L_{n}^{[2]} + 2 L_{n - 1}^{[2]}}{5}$
3	$(\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{array})$	$(\begin{array}{rrr} \frac{1}{22} & \frac{9}{22} & - \frac{2}{11} \\ - \frac{2}{11} & - \frac{3}{22} & \frac{5}{22} \\ \frac{5}{22} & \frac{1}{22} & \frac{1}{11} \end{array})$	$\frac{2 L_{n}^{[3]} + L_{n - 1}^{[3]} + 5 L_{n - 2}^{[3]}}{22}$
4	$(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \end{array})$	$(\begin{array}{rrrr} \frac{10}{563} & \frac{157}{563} & - \frac{103}{563} & \frac{16}{563} \\ \frac{16}{563} & \frac{26}{563} & \frac{173}{563} & - \frac{87}{563} \\ - \frac{87}{563} & - \frac{71}{563} & - \frac{61}{563} & \frac{86}{563} \\ \frac{86}{563} & - \frac{1}{563} & \frac{15}{563} & \frac{25}{563} \end{array})$	$\frac{25 L_{n}^{[4]} + 15 L_{n - 1}^{[4]} - L_{n - 2}^{[4]} + 86 L_{n - 3}^{[4]}}{563}$
5	$(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{array})$	$(\begin{array}{rrrrr} \frac{9}{1198} & \frac{127}{599} & - \frac{99}{599} & \frac{5}{599} & \frac{15}{1198} \\ \frac{15}{1198} & \frac{12}{599} & \frac{269}{1198} & - \frac{183}{1198} & \frac{25}{1198} \\ \frac{25}{1198} & \frac{20}{599} & \frac{49}{1198} & \frac{147}{599} & - \frac{79}{599} \\ - \frac{79}{599} & - \frac{133}{1198} & - \frac{59}{599} & - \frac{109}{1198} & \frac{68}{599} \\ \frac{68}{599} & - \frac{11}{599} & \frac{3}{1198} & \frac{9}{599} & \frac{27}{1198} \end{array})$	$\frac{27 L_{n}^{[5]} + 18 L_{n - 1}^{[5]} + 3 L_{n - 2}^{[5]} - 22 L_{n - 3}^{[5]} + 136 L_{n - 4}^{[5]}}{1198}$
6	$(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 \end{array})$	$(\begin{array}{rrrrrr} \frac{686}{205937} & \frac{35409}{205937} & - \frac{30143}{205937} & \frac{264}{205937} & \frac{840}{205937} & \frac{1176}{205937} \\ \frac{1176}{205937} & \frac{1862}{205937} & \frac{36585}{205937} & - \frac{28967}{205937} & \frac{1440}{205937} & \frac{2016}{205937} \\ \frac{2016}{205937} & \frac{3192}{205937} & \frac{3878}{205937} & \frac{38601}{205937} & - \frac{26951}{205937} & \frac{3456}{205937} \\ \frac{3456}{205937} & \frac{5472}{205937} & \frac{6648}{205937} & \frac{7334}{205937} & \frac{42057}{205937} & - \frac{23495}{205937} \\ - \frac{23495}{205937} & - \frac{20039}{205937} & - \frac{18023}{205937} & - \frac{16847}{205937} & - \frac{16161}{205937} & \frac{18562}{205937} \\ \frac{18562}{205937} & - \frac{4933}{205937} & - \frac{1477}{205937} & \frac{539}{205937} & \frac{1715}{205937} & \frac{2401}{205937} \end{array})$	$\frac{2401 L_{n}^{[6]} + 1715 L_{n - 1}^{[6]} + 539 L_{n - 2}^{[6]} - 1477 L_{n - 3}^{[6]} - 4933 L_{n - 4}^{[6]} + 18562 L_{n - 5}^{[6]}}{205937}$