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

%I #9 Dec 22 2016 23:19:34

%S 0,0,0,8,24,144,240,672,960,2000,2640,4680,5880,9408,11424,17024,

%T 20160,28512,33120,45000,51480,67760,76560,98208,109824,137904,152880,

%U 188552,207480,252000,275520,330240,359040,425408,460224,539784,581400,675792,725040,836000,893760,1023120,1090320,1240008

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

%H Indranil Ghosh, <a href="/A279483/b279483.txt">Table of n, a(n) for n = 0..200</a>

%H <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 From _Colin Barker_, Dec 13 2016: (Start)

%F a(n) = (3*n^4 - 8*n^3 - 12*n^2 + 32*n)/8 for n even.

%F a(n) = (3*n^4 - 4*n^3 - 10*n^2 + 4*n + 7)/8 for n odd.

%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>8.

%F G.f.: 8*x^3*(1 + 2*x + 11*x^2 + 4*x^3) / ((1 - x)^5*(1 + x)^4).

%F (End)

%t CoefficientList[Series[8 x^3*(1 + 2 x + 11 x^2 + 4 x^3)/((1 - x)^5*(1 + x)^4), {x, 0, 43}], x] (* _Michael De Vlieger_, Dec 13 2016 *)

%o (Python)

%o def t(n):

%o s=0

%o for a in range(0,n+1):

%o for b in range(0,n+1):

%o for c in range(0,n+1):

%o for d in range(0,n+1):

%o if (a!=b and a!=d and b!=d and c!=a and c!=b and c!=d):

%o if (a*d-b*c)%2==1:

%o s+=1

%o return s

%o for i in range(0,201):

%o print str(i)+" "+str(t(i))

%o (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)} \\ a(n)=F(n, 0)[2];

%o (PARI) concat(vector(3), Vec(8*x^3*(1 + 2*x + 11*x^2 + 4*x^3) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ _Colin Barker_, Dec 13 2016

%Y Cf. A210370 (where the entries can be repeated).

%K nonn,easy

%O 0,4

%A _Indranil Ghosh_, Dec 13 2016