A176233 Determinant of n X n matrix with rows (n^2,-1,0,...,0), (1,n^2,-1, 0,...,0), (0,1,n^2,-1,0,...,0), ...,(0,0,...,1,n^2).

1, 17, 747, 66305, 9828200, 2185188193, 679919101029, 281956264747009, 150277722869740455, 100090028003500150001, 81458362232421250207824, 79539026883848399173231873, 91771878445323959814042316673

Offset: 1

Author

Michel Lagneau, Apr 12 2010

Keywords

• nonn

Comments

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).

Examples

			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].
		

References

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

Crossrefs

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

Programs

Maple

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 :

Mathematica

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 *)

Formula

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