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COMMENTS
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Determinant of n+1 X n+1 matrix: ((F(0),-1,0,...,0),(1,F(1),-1,0,...,0),(0,1,F(2),-1,0,...,0),...,(0,0,...,1,F(n)). Each determinant is the numerator of the fraction x(n)/y(n) equal to the continued fraction expansion of the diagonal elements [F(0), F(1), ..., F(n)] of the n+1 X n+1 matrix. The value x(n) is obtained by computing the determinant det(n+1 X n+1) from the last column. The value y(n) is obtained by computing this determinant after removal of the first row and the first column (see example below).
The sequence A001040 give the values of each determinant with numerator of continued fraction given by the expansion of the diagonal elements [n,n-1,...,3,2,1]. The same is true for the sequence A084845 with the expansion of the diagonal elements [n,n,...,n], and the sequence A036246 for the elements[ 0, 1, 4, ..., n^2 ].
Examples:
for n = 0, det[0] = 0; for n = 1, det(([[0,-1],[1,1]]) = 1;
for n = 2, det([[0,-1, 0],[1,1,-1],[0,1,1]]))=1;
for n = 3, det([[0,-1, 0,0],[1,1,-1,0],[0,1,1,-1],[0,0,1,2]])) = 3, and the continued fraction expansion is 3/det(([[1,-1, 0],[1,1,-1],[0,1,2]])) = 5/3 = 0 + 1 + 1/(1 + 1/2) => [0,1,1,2]. (End)
a(n) is the denominator of the continued fraction [F(1), F(2), ..., F(n)] for n > 0. - Seung Ju Lee, Aug 23 2020
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