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A106634 Number of ways to express n as i*j+k*l, with i,j,k,l in the range [0..n].

%I A106634 #26 Jul 21 2024 12:47:54
%S A106634 1,6,21,32,62,58,124,88,173,158,226,156,380,194,340,356,466,274,613,
%T A106634 316,690,536,596,404,1060,552,734,728,1032,546,1376,596,1213,932,1026,
%U A106634 976,1858,750,1180,1144,1910,854,2048,908,1784,1730,1500,1016,2800
%N A106634 Number of ways to express n as i*j+k*l, with i,j,k,l in the range [0..n].
%C A106634 Number of ordered 4-tuples [i,j,k,l] with n=i*j+k*l and i,j,k,l in the range [0..n].
%C A106634 a(n) is odd iff n is in A001105.
%H A106634 R. J. Mathar and Charles R Greathouse IV, <a href="/A106634/b106634.txt">Table of n, a(n) for n = 0..10000</a> (terms through 780 from Mathar)
%F A106634 From _Ridouane Oudra_, Jul 20 2024: (Start)
%F A106634 a(n) = (4*n + 2)*tau(n) + Sum_{i=1..n-1} tau(i)*tau(n-i), for n>0 ;
%F A106634 a(n) = (4*n + 2)*A000005(n) + A055507(n-1), for n>0 ;
%F A106634 a(n) = 4*A038040(n) + A062011(n) + A055507(n-1), for n>0. (End)
%e A106634 a(1)=6: the 4-tuples ijkl are 1100, 1101, 1110, 0011, 0111, 1011.
%e A106634 a(2)=21: 1111, 2100, 210x, 21x0, 1200, 120x, 12x0, where x = 1 or 2, and ten more with the two halves swapped.
%p A106634 A106634 := proc(n)
%p A106634     local a,i,j,k,l ;
%p A106634     a := 0 ;
%p A106634     for i from 0 to n do
%p A106634         for j from 0 to n do
%p A106634             if i*j > n then
%p A106634                 break;
%p A106634             end if;
%p A106634             for k from 0 to n do
%p A106634                 if i*j = n then
%p A106634                     # treat l=0 separately
%p A106634                     a := a+1 ;
%p A106634                 end if;
%p A106634                 # l=1..n
%p A106634                 if k =0 then
%p A106634                     if i*j=n then
%p A106634                         a := a+n ;
%p A106634                     end if;
%p A106634                 else
%p A106634                     l := (n-i*j)/k ;
%p A106634                     if l >=1 and l <=n and type(l,'integer') then
%p A106634                         a := a+1 ;
%p A106634                     end if;
%p A106634                 end if;
%p A106634             end do:
%p A106634         end do:
%p A106634     end do:
%p A106634     a ;
%p A106634 end proc: # _R. J. Mathar_, Oct 17 2012
%t A106634 list[n_] := Module[{v, i, j}, v[_] = 0;
%t A106634 For[i = 2, i <= n, i++, For[j = 1, j <= Min[Quotient[n, i], i-1], j++, v[i*j]+= 2]];
%t A106634 For[i = 1, i <= Floor@Sqrt[n], i++, v[i^2]++];
%t A106634 Join[{1}, Table[2 Sum[v[j] v[i-j], {j, Quotient[i, 2]+1, i-1}]+If[OddQ[i], 0, v[i/2]^2]+(4i+2) v[i], {i, 1, n}]]];
%t A106634 list[48] (* _Jean-François Alcover_, Jun 04 2023, after _Charles R Greathouse IV_ *)
%o A106634 (PARI) list(n)={
%o A106634     my(v=vector(n));
%o A106634     for(i=2,n,for(j=1,min(n\i,i-1),v[i*j]+=2));
%o A106634     for(i=1,sqrtint(n),v[i^2]++);
%o A106634     concat(1,vector(n,i,2*sum(j=i\2+1,i-1,v[j]*v[i-j])+if(i%2,,v[i/2]^2)+(4*i+2)*v[i]))
%o A106634 }; \\ _Charles R Greathouse IV_, Oct 17 2012
%Y A106634 Cf. A001105, A055507, A106633, A106846, A106847.
%Y A106634 Cf. A000005, A038040, A062011.
%K A106634 nonn,easy
%O A106634 0,2
%A A106634 _Ralf Stephan_, May 06 2005
%E A106634 Definition clarified by _N. J. A. Sloane_, Jul 07 2012