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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).
3
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.
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Apr 12 2010
STATUS
approved