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A279483
Number of 2 X 2 matrices with entries in {0,1,...,n} and odd determinant with no entry repeated.
1
0, 0, 0, 8, 24, 144, 240, 672, 960, 2000, 2640, 4680, 5880, 9408, 11424, 17024, 20160, 28512, 33120, 45000, 51480, 67760, 76560, 98208, 109824, 137904, 152880, 188552, 207480, 252000, 275520, 330240, 359040, 425408, 460224, 539784, 581400, 675792, 725040, 836000, 893760, 1023120, 1090320, 1240008
OFFSET
0,4
FORMULA
From Colin Barker, Dec 13 2016: (Start)
a(n) = (3*n^4 - 8*n^3 - 12*n^2 + 32*n)/8 for n even.
a(n) = (3*n^4 - 4*n^3 - 10*n^2 + 4*n + 7)/8 for n odd.
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.
G.f.: 8*x^3*(1 + 2*x + 11*x^2 + 4*x^3) / ((1 - x)^5*(1 + x)^4).
(End)
MATHEMATICA
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 *)
PROG
(Python)
def t(n):
s=0
for a in range(0, n+1):
for b in range(0, n+1):
for c in range(0, n+1):
for d in range(0, n+1):
if (a!=b and a!=d and b!=d and c!=a and c!=b and c!=d):
if (a*d-b*c)%2==1:
s+=1
return s
for i in range(0, 201):
print str(i)+" "+str(t(i))
(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];
(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
CROSSREFS
Cf. A210370 (where the entries can be repeated).
Sequence in context: A305224 A182068 A092771 * A098070 A100042 A061027
KEYWORD
nonn,easy
AUTHOR
Indranil Ghosh, Dec 13 2016
STATUS
approved