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A210369 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even determinant.

%I A210369 #28 Jul 16 2024 13:07:02
%S A210369 1,10,65,160,457,810,1681,2560,4481,6250,9841,12960,18985,24010,33377,
%T A210369 40960,54721,65610,84961,100000,126281,146410,181105,207360,252097,
%U A210369 285610,342161,384160,454441,506250,592321,655360,759425,835210,959617,1049760
%N A210369 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even determinant.
%C A210369 a(n) is also the number of 2 X 2 matrices with all terms in {0,1,...n} and even permanent.
%C A210369 The determinant will be even if either all entries are odd or if both the leading and trailing diagonals have no more than one odd entry each. - _Andrew Howroyd_, Apr 28 2020
%H A210369 Andrew Howroyd, <a href="/A210369/b210369.txt">Table of n, a(n) for n = 0..1000</a>
%H A210369 <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 A210369 a(n) + A210370(n) = n^4.
%F A210369 From _Colin Barker_, Nov 28 2014: (Start)
%F A210369 a(n) = (13 + 3*(-1)^n + 4*(13+3*(-1)^n)*n + 2*(37+7*(-1)^n)*n^2 + 4*(11+(-1)^n)*n^3 + 10*n^4)/16.
%F A210369 G.f.: -(x^7+9*x^6+27*x^5+83*x^4+59*x^3+51*x^2+9*x+1) / ((x-1)^5*(x+1)^4).
%F A210369 (End)
%F A210369 a(n) = ((n+1)^2 - ceiling(n/2)^2)^2 + ceiling(n/2)^4. - _Andrew Howroyd_, Apr 28 2020
%t A210369 a = 0; b = n; z1 = 28;
%t A210369 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A210369 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A210369 u[n_] := Sum[c[n, 2 k], {k, -n^2, n^2}]
%t A210369 v[n_] := Sum[c[n, 2 k - 1], {k, -n^2, n^2}]
%t A210369 Table[u[n], {n, 0,  z1}] (* A210369 *)
%t A210369 Table[v[n], {n, 0, z1}]  (* A210370 *)
%o A210369 (PARI) a(n) = {((n+1)^2 - ceil(n/2)^2)^2 + ceil(n/2)^4} \\ _Andrew Howroyd_, Apr 28 2020
%Y A210369 Cf. A210000, A210370.
%K A210369 nonn
%O A210369 0,2
%A A210369 _Clark Kimberling_, Mar 20 2012
%E A210369 Terms a(29) and beyond from _Andrew Howroyd_, Apr 28 2020