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Coefficients of 子葉さん's k-Fibonacci formula

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

$F_n^{[k]}=\frac{k-1}{2(2k)^k-(k+1)^{k+1}}\left(\sum^{k-1}_{j=1}(k+1)^{k-1-j}((k+1)^j-2(2k)^{j-1})L_{n-j+1}^{[k]}+\frac{2((2k)^{k-1}-(k+1)^{k-1})}{k-1}L_{n-k+1}^{[k]}\right)$

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
$$ M = \begin{pmatrix} & 1 & & \\ & & \ddots & \\ & & & 1 \\ 1 & \cdots & \cdots & 1 \end{pmatrix} $$

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

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