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A278846 Number of unimodular 2 X 2 matrices having entries in {0,1,...,n} with no entry repeated.

%I A278846 #38 Jun 11 2024 23:46:16
%S A278846 0,0,0,0,0,8,8,40,48,80,88,152,160,232,264,304,344,448,480,608,648,
%T A278846 720,784,944,968,1104,1176,1304,1376,1576,1616,1840,1944,2080,2184,
%U A278846 2352,2424,2688,2816,2984,3072,3368,3440,3760,3896,4064,4224,4576,4664,4984,5120
%N A278846 Number of unimodular 2 X 2 matrices having entries in {0,1,...,n} with no entry repeated.
%C A278846 a(n) mod 8 = 0.
%H A278846 Robert Israel, <a href="/A278846/b278846.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..241 from Indranil Ghosh)
%p A278846 df:= proc(n) local count, c,d,q,av,bc,a,b;
%p A278846   count:= 0:
%p A278846   for d from 1 to n-1 do
%p A278846       av:= {$1..n-1} minus {d};
%p A278846       for q in [-1,1] do
%p A278846         bc:= n*d+q;
%p A278846         for b in numtheory:-divisors(bc) intersect av do
%p A278846           c:= bc/b;
%p A278846           if c < b and member(c,av) then count:=count+8 fi;
%p A278846   od od od;
%p A278846   count
%p A278846 end proc:
%p A278846 ListTools:-PartialSums(map(df, [$0..100])); # _Robert Israel_, Nov 29 2016
%t A278846 df[n_] := Module[{count = 0, c, d, q, av, bc, a, b}, Do[av = Range[n - 1]  ~Complement~ {d}; Do[bc = n d + q; Do[c = bc/b; If[c < b && MemberQ[av, c], count += 8], {b, Divisors[bc] ~Intersection~ av}], {q, {-1 , 1}}], {d, 1, n - 1}]; count];
%t A278846 df /@ Range[0, 100] // Accumulate (* _Jean-François Alcover_, Jul 29 2020, after _Robert Israel_ *)
%o A278846 (Python)
%o A278846 def a(n):
%o A278846     s=0
%o A278846     for a in range(0,n+1):
%o A278846         for b in range(0,n+1):
%o A278846             for c in range(0,n+1):
%o A278846                 for d in range(0,n+1):
%o A278846                     if (a!=b  and a!=d and b!=d and c!=a and c!=b and c!=d):
%o A278846                         if abs(a*d-b*c)==1:
%o A278846                             s+=1
%o A278846     return s
%o A278846 print([a(n) for n in range(0, 52)]) # _Indranil Ghosh_, Nov 29 2016
%Y A278846 Cf. A210000 (where the matrix entries can be repeated).
%K A278846 nonn
%O A278846 0,6
%A A278846 _Indranil Ghosh_, Nov 29 2016