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

1, 6, 21, 32, 62, 58, 124, 88, 173, 158, 226, 156, 380, 194, 340, 356, 466, 274, 613, 316, 690, 536, 596, 404, 1060, 552, 734, 728, 1032, 546, 1376, 596, 1213, 932, 1026, 976, 1858, 750, 1180, 1144, 1910, 854, 2048, 908, 1784, 1730, 1500, 1016, 2800

Offset: 0

Author

Ralf Stephan, May 06 2005

Keywords

• nonn
• easy

Comments

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].

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

Examples

			a(1)=6: the 4-tuples ijkl are 1100, 1101, 1110, 0011, 0111, 1011.
a(2)=21: 1111, 2100, 210x, 21x0, 1200, 120x, 12x0, where x = 1 or 2, and ten more with the two halves swapped.
		

Links

• R. J. Mathar and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms through 780 from Mathar)

Crossrefs

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

Cf. A000005, A038040, A062011.

Programs

Maple

A106634 := proc(n)
    local a,i,j,k,l ;
    a := 0 ;
    for i from 0 to n do
        for j from 0 to n do
            if i*j > n then
                break;
            end if;
            for k from 0 to n do
                if i*j = n then
                    # treat l=0 separately
                    a := a+1 ;
                end if;
                # l=1..n
                if k =0 then
                    if i*j=n then
                        a := a+n ;
                    end if;
                else
                    l := (n-i*j)/k ;
                    if l >=1 and l <=n and type(l,'integer') then
                        a := a+1 ;
                    end if;
                end if;
            end do:
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

Mathematica

list[n_] := Module[{v, i, j}, v[_] = 0;
For[i = 2, i <= n, i++, For[j = 1, j <= Min[Quotient[n, i], i-1], j++, v[i*j]+= 2]];
For[i = 1, i <= Floor@Sqrt[n], i++, v[i^2]++];
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}]]];
list[48] (* Jean-François Alcover, Jun 04 2023, after Charles R Greathouse IV *)

PARI

list(n)={
    my(v=vector(n));
    for(i=2,n,for(j=1,min(n\i,i-1),v[i*j]+=2));
    for(i=1,sqrtint(n),v[i^2]++);
    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]))
}; \\ Charles R Greathouse IV, Oct 17 2012

Formula

From Ridouane Oudra, Jul 20 2024: (Start)

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

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

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

Extensions

Definition clarified by N. J. A. Sloane, Jul 07 2012