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A210370 Number of 2 X 2 matrices with all elements in {0,1,...,n} and odd determinant.

%I A210370 #22 Jul 16 2024 13:08:49
%S A210370 0,6,16,96,168,486,720,1536,2080,3750,4800,7776,9576,14406,17248,
%T A210370 24576,28800,39366,45360,60000,68200,87846,98736,124416,138528,171366,
%U A210370 189280,230496,252840,303750,331200,393216,426496,501126,541008,629856,677160,781926,837520
%N A210370 Number of 2 X 2 matrices with all elements in {0,1,...,n} and odd determinant.
%C A210370 a(n) is also the number of 2 X 2 matrices with all elements in {0,1,...n} and odd permanent.
%H A210370 Andrew Howroyd, <a href="/A210370/b210370.txt">Table of n, a(n) for n = 0..1000</a>
%H A210370 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 4, -4, -6, 6, 4, -4, -1, 1).
%F A210370 a(n) + A210369(n) = n^4.
%F A210370 From _Colin Barker_, Nov 28 2014: (Start)
%F A210370 a(n) = (3 - 3*(-1)^n - 12*(-1+(-1)^n)*n + (22-14*(-1)^n)*n^2 - 4*(-5+(-1)^n)*n^3 + 6*n^4)/16.
%F A210370 G.f.: -2*x*(3*x^5+17*x^4+16*x^3+28*x^2+5*x+3) / ((x-1)^5*(x+1)^4).
%F A210370 (End)
%F A210370 a(n) = 2*((n+1)^2 - ceiling(n/2)^2)*ceiling(n/2)^2. - _Andrew Howroyd_, Apr 28 2020
%t A210370 a = 0; b = n; z1 = 28;
%t A210370 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210370 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210370 u[n_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
%t A210370 v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
%t A210370 Table[u[n], {n, 0,  z1}] (* A210369 *)
%t A210370 Table[v[n], {n, 0, z1}](* A210370 *)
%o A210370 (PARI) a(n)={2*((n+1)^2-ceil(n/2)^2)*ceil(n/2)^2} \\ _Andrew Howroyd_, Apr 28 2020
%Y A210370 Cf. A210000, A210369.
%K A210370 nonn
%O A210370 0,2
%A A210370 _Clark Kimberling_, Mar 20 2012
%E A210370 Terms a(29) and beyond from _Andrew Howroyd_, Apr 28 2020