1,2

Each determinant is the numerator of the fraction x(n)/y(n) = [n^2, n^2, ..., n^2] (simple continued fraction). The value x(n) is obtained by computing the determinant det(n X n) along 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).

J. M. De Koninck and A. Mercier, 1001 problèmes en théorie classique des nombres. Collection ellipses (2004), p. 115.

a(n) ~ n^(2*n). - Vaclav Kotesovec, Dec 29 2019

a(n) = ( ((n^2 + sqrt(n^4+4))/2)^(n+1) - ((n^2 - sqrt(n^4+4))/2)^(n+1) ) / sqrt(n^4+4). - Sela Fried, Feb 01 2026

For n = 1, det[1] = 1.

For n = 2, det([[4,-1],[1,4]]) = 17, and the continued fraction expansion is 17/4 = [2^2,2^2].

For n = 3, det([[9,-1,0],[1,9,-1],[0,1,9]]) = 747, and the continued fraction expansion is 747/det([[9,-1],[1,9]]) = 747/82 = [3^2,3^2,3^2].

for n from 15 by -1 to 1 do x0:=n^2: for p from n by -1 to 2 do : x0:= n^2 + 1/x0 :od: print(x0): od :

nmax = 20; Do[x0 = n^2; Do[x0 = n^2 + 1/x0, {p, n, 2, -1}]; a[n] = Numerator[x0]; , {n, nmax, 1, -1}]; Table[a[n], {n, 1, nmax}] (* Vaclav Kotesovec, Dec 29 2019 *)

Cf. A001040, A036245, A036246, A084845, A176232.

Sequence in context: A171766 A283579 A294757 * A360647 A355495 A368492

Adjacent sequences: A176230 A176231 A176232 * A176234 A176235 A176236

Michel Lagneau, Apr 12 2010

More terms from Michel Marcus, Feb 01 2026

approved