cp's OEIS Frontend

A279483 Number of 2 X 2 matrices with entries in {0,1,...,n} and odd determinant with no entry repeated.

%I A279483 #12 Aug 13 2025 21:27:24
%S A279483 0,0,0,8,24,144,240,672,960,2000,2640,4680,5880,9408,11424,17024,
%T A279483 20160,28512,33120,45000,51480,67760,76560,98208,109824,137904,152880,
%U A279483 188552,207480,252000,275520,330240,359040,425408,460224,539784,581400,675792,725040,836000,893760,1023120,1090320,1240008
%N A279483 Number of 2 X 2 matrices with entries in {0,1,...,n} and odd determinant with no entry repeated.
%H A279483 Indranil Ghosh, <a href="/A279483/b279483.txt">Table of n, a(n) for n = 0..200</a>
%H A279483 <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 A279483 From _Colin Barker_, Dec 13 2016: (Start)
%F A279483 a(n) = (3*n^4 - 8*n^3 - 12*n^2 + 32*n)/8 for n even.
%F A279483 a(n) = (3*n^4 - 4*n^3 - 10*n^2 + 4*n + 7)/8 for n odd.
%F A279483 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 A279483 G.f.: 8*x^3*(1 + 2*x + 11*x^2 + 4*x^3) / ((1 - x)^5*(1 + x)^4).
%F A279483 (End)
%t A279483 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 A279483 (Python)
%o A279483 def t(n):
%o A279483     s=0
%o A279483     for a in range(0,n+1):
%o A279483         for b in range(0,n+1):
%o A279483             for c in range(0,n+1):
%o A279483                 for d in range(0,n+1):
%o A279483                    if (a!=b  and a!=d and b!=d and c!=a and c!=b and c!=d):
%o A279483                         if (a*d-b*c)%2==1:
%o A279483                             s+=1
%o A279483     return s
%o A279483 for i in range(0,201):
%o A279483     print(i, t(i))
%o A279483 (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 A279483 (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 A279483 Cf. A210370 (where the entries can be repeated).
%K A279483 nonn,easy
%O A279483 0,4
%A A279483 _Indranil Ghosh_, Dec 13 2016