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

1,2

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).

References

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

Links

Table of n, a(n) for n=1..13.

Formula

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

Example

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].

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 *)

Crossrefs

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

Sequence in context: A171766 A283579 A294757 * A360647 A355495 A368492

Adjacent sequences: A176230 A176231 A176232 * A176234 A176235 A176236

Keyword

nonn

Author

Michel Lagneau, Apr 12 2010

Status

approved