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A277044 Number of 2 X 2 matrices with entries in {0,1,...,n} and even determinant with no entry repeated.

%I A277044 #30 May 19 2025 23:45:03
%S A277044 0,0,0,16,96,216,600,1008,2064,3040,5280,7200,11280,14616,21336,26656,
%T A277044 36960,44928,59904,71280,92160,107800,135960,156816,193776,220896,
%U A277044 268320,302848,362544,405720,479640,532800,623040,687616,796416,873936,1003680,1095768,1248984,1357360,1536720,1663200
%N A277044 Number of 2 X 2 matrices with entries in {0,1,...,n} and even determinant with no entry repeated.
%C A277044 a(n) mod 8 = 0.
%H A277044 Indranil Ghosh, <a href="/A277044/b277044.txt">Table of n, a(n) for n = 0..200</a>
%H A277044 <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 A277044 From _Colin Barker_ and _Charles R Greathouse IV_, Dec 12 2016: (Start)
%F A277044 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>8.
%F A277044 a(n) = (5*n^4 - 8*n^3 + 4*n^2 - 16*n)/8 for n even.
%F A277044 a(n) = (5*n^4 - 12*n^3 + 2*n^2 + 12*n - 7)/8 for n odd.
%F A277044 G.f.: 8*x^3*(2 + 10*x + 7*x^2 + 8*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^4).
%F A277044 (End)
%o A277044 (Python)
%o A277044 def t(n):
%o A277044     s=0
%o A277044     for a in range(0,n+1):
%o A277044         for b in range(0,n+1):
%o A277044             for c in range(0,n+1):
%o A277044                 for d in range(0,n+1):
%o A277044                    if (a!=b  and a!=d and b!=d and c!=a and c!=b and c!=d):
%o A277044                         if (a*d-b*c)%2==0:
%o A277044                             s+=1
%o A277044     return s
%o A277044 for i in range(0,201):
%o A277044     print(f"{i} {t(i)}")
%o A277044 (PARI) F(n,{r=0})={my(s=vector(2),v);forvec(y=vector(4,j,[0,n]),for(k=23*!!r,23,v=numtoperm(4,k);s[1+(y[v[1]]*y[v[4]]-y[v[3]]*y[v[2]])%2]++),2*!r);return(s)} \\ Use r=1 for A210369;
%o A277044 a(n)=F(n,0)[1]; \\ Also works for A210370 if F(n,1)[2] is used instead. - _R. J. Cano_, Dec 12 2016
%o A277044 (PARI) a(n)=my(e=n\2+1,o=(n+1)\2); 24*binomial(o,4) + 16*binomial(e,2)*binomial(o,2) + 24*o*binomial(e,3) + 24*binomial(e,4) \\ _Charles R Greathouse IV_, Dec 12 2016
%Y A277044 Cf. A210369 (where the entries can be repeated).
%K A277044 nonn,easy
%O A277044 0,4
%A A277044 _Indranil Ghosh_, Dec 12 2016