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%I #17 May 05 2020 17:20:51
%S 1,41,457,1345,4481,8521,18985,30017,54721,78121,126281,168961,252097,
%T 322505,454441,562561,759425,916777,1197001,1416641,1800961,2097481,
%U 2608937,2998465,3662401,4162601,5006665,5636737,6690881,7471561,8768041,9721601,11294977,12445225,14332361,15704641
%N Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and even determinant.
%C A211154(n) + A211155(n) = (2n+1)^4.
%C For a guide to related sequences, see A210000.
%H Andrew Howroyd, <a href="/A211154/b211154.txt">Table of n, a(n) for n = 0..1000</a>
%F From _Chai Wah Wu_, Nov 27 2016: (Start)
%F a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
%F G.f.: x*(-x^8 - 36*x^6 - 416*x^5 - 734*x^4 - 1472*x^3 - 724*x^2 - 416*x - 41)/((x - 1)^5*(x + 1)^4). (End)
%p seq((2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n), n=1..20); # _Mark van Hoeij_, May 13 2013
%t a = -n; b = n; z1 = 20;
%t t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t c[n_, k_] := c[n, k] = Count[t[n], k]
%t u[n_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
%t v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
%t Table[u[n], {n, 1, z1}] (* A211154 *)
%t Table[v[n], {n, 1, z1}] (* A211155 *)
%o (PARI) a(n)=(2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n); \\ _Joerg Arndt_, May 14 2013
%Y Cf. A210000, A211155.
%K nonn
%O 0,2
%A _Clark Kimberling_, Apr 05 2012
%E More terms from _Joerg Arndt_, May 14 2013
%E a(0)=1 prepended by _Andrew Howroyd_, May 05 2020
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