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%I #20 Jan 06 2017 16:01:45
%S 1,5,25,101,385,1397,4921,16949,57409,191909,634777,2081477,6775873,
%T 21921941,70548793,225995285,721032577,2292237893,7264134169,
%U 22954663973,72350776321,227512682165,713919106105,2235900497141,6990131027905,21817681693157
%N Absolute value of determinant of n X n matrix where the element a(i,j) = if i + j > n then 2*(i + j -n) - 1, else 2*(n + 1 - i - j).
%C The main antidiagonal is 1, the upper left elements are increasing larger even numbers and the lower right elements are increasing larger odd numbers.
%H Harvey P. Dale, <a href="/A083877/b083877.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-9).
%F a(n) = 1/12 * [(4n-1)3^n - 3(-1)^n].
%F a(n) = 5*a(n-1)-3*a(n-2)-9*a(n-3). G.f.: x*(3*x^2+1) / ((x+1)*(3*x-1)^2). - _Colin Barker_, Sep 28 2014
%e a(5) = det{ 8 6 4 2 1 / 6 4 2 1 3 / 4 2 1 3 5 / 2 1 3 5 7 / 1 3 5 7 9 } = 385.
%t f[i_, j_, n_] := Block[{a = 2*(i + j) - 2*n - 1}, If[i + j <= n, a = Abs[a - 1]]; a]; Table[ Abs[ Det[ Table[ f[i, j, n], {i, 1, n}, {j, 1, n}]]], {n, 1, 24}]
%t LinearRecurrence[{5,-3,-9},{1,5,25},30] (* _Harvey P. Dale_, Jan 06 2017 *)
%o (PARI) Vec(x*(3*x^2+1)/((x+1)*(3*x-1)^2) + O(x^100)) \\ _Colin Barker_, Sep 28 2014
%o (PARI) a(n) = abs(matdet(matrix (n, n, i, j, if (i + j > n, 2*(i + j -n) - 1, 2*(n + 1 - i - j))))); \\ _Michel Marcus_, Sep 29 2014
%K nonn,easy
%O 1,2
%A _Robert G. Wilson v_, May 07 2003
%E More terms from _Colin Barker_, Sep 28 2014
%E Definition corrected by _Michel Marcus_, Oct 01 2014