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Determinant of the n X n matrix whose element (i,j) equals |i-j| (Mod 3).
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%I #20 Jun 13 2015 00:50:33

%S 0,-1,4,9,-10,4,18,-19,4,27,-28,4,36,-37,4,45,-46,4,54,-55,4,63,-64,4,

%T 72,-73,4,81,-82,4,90,-91,4,99,-100,4,108,-109,4,117,-118,4,126,-127,

%U 4,135,-136,4,144,-145,4,153,-154,4,162,-163,4,171,-172,4,180,-181,4,189

%N Determinant of the n X n matrix whose element (i,j) equals |i-j| (Mod 3).

%C (Mod 2) for n > 2 produces nothing but zeros.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,1,1,1).

%F a(3k) = 4, a(3k+1) = 9*k, a(3k+2) = -9*k-1.

%F a(n) = -a(n-1)-a(n-2)+a(n-3)+a(n-4)+a(n-5). - _Colin Barker_, Sep 29 2014

%F G.f.: -x^2*(2*x+1)*(2*x^2+5*x-1) / ((x-1)*(x^2+x+1)^2). - _Colin Barker_, Sep 29 2014

%t Table[ Det[ Table[ Mod[ Abs[i - j], 3], {i, 1, n}, {j, 1, n}]], {n, 1, 65}]

%o (PARI) a(n) = matdet(matrix (n, n, i, j, abs(i-j) % 3)); \\ _Michel Marcus_, Sep 29 2014

%o (PARI) concat(0, Vec(-x^2*(2*x+1)*(2*x^2+5*x-1)/((x-1)*(x^2+x+1)^2) + O(x^100))) \\ _Colin Barker_, Sep 30 2014

%Y Cf. A071769 (with Mod 4).

%K sign,easy

%O 1,3

%A _Robert G. Wilson v_, Jun 04 2002