A106634 - OEIS

%I #26 Jul 21 2024 12:47:54

%S 1,6,21,32,62,58,124,88,173,158,226,156,380,194,340,356,466,274,613,

%T 316,690,536,596,404,1060,552,734,728,1032,546,1376,596,1213,932,1026,

%U 976,1858,750,1180,1144,1910,854,2048,908,1784,1730,1500,1016,2800

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

%C 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 a(n) is odd iff n is in A001105.

%H 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 From _Ridouane Oudra_, Jul 20 2024: (Start)

%F a(n) = (4*n + 2)*tau(n) + Sum_{i=1..n-1} tau(i)*tau(n-i), for n>0 ;

%F a(n) = (4*n + 2)*A000005(n) + A055507(n-1), for n>0 ;

%F a(n) = 4*A038040(n) + A062011(n) + A055507(n-1), for n>0. (End)

%e a(1)=6: the 4-tuples ijkl are 1100, 1101, 1110, 0011, 0111, 1011.

%e 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 := proc(n)

%p     local a,i,j,k,l ;

%p     a := 0 ;

%p     for i from 0 to n do

%p         for j from 0 to n do

%p             if i*j > n then

%p                 break;

%p             end if;

%p             for k from 0 to n do

%p                 if i*j = n then

%p                     # treat l=0 separately

%p                     a := a+1 ;

%p                 end if;

%p                 # l=1..n

%p                 if k =0 then

%p                     if i*j=n then

%p                         a := a+n ;

%p                     end if;

%p                 else

%p                     l := (n-i*j)/k ;

%p                     if l >=1 and l <=n and type(l,'integer') then

%p                         a := a+1 ;

%p                     end if;

%p                 end if;

%p             end do:

%p         end do:

%p     end do:

%p     a ;

%p end proc: # _R. J. Mathar_, Oct 17 2012

%t list[n_] := Module[{v, i, j}, v[_] = 0;

%t For[i = 2, i <= n, i++, For[j = 1, j <= Min[Quotient[n, i], i-1], j++, v[i*j]+= 2]];

%t For[i = 1, i <= Floor@Sqrt[n], i++, v[i^2]++];

%t 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 list[48] (* _Jean-François Alcover_, Jun 04 2023, after _Charles R Greathouse IV_ *)

%o (PARI) list(n)={

%o     my(v=vector(n));

%o     for(i=2,n,for(j=1,min(n\i,i-1),v[i*j]+=2));

%o     for(i=1,sqrtint(n),v[i^2]++);

%o     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 }; \\ _Charles R Greathouse IV_, Oct 17 2012

%Y Cf. A001105, A055507, A106633, A106846, A106847.

%Y Cf. A000005, A038040, A062011.

%K nonn,easy

%O 0,2

%A _Ralf Stephan_, May 06 2005

%E Definition clarified by _N. J. A. Sloane_, Jul 07 2012