A106633 Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].

1, 4, 8, 12, 17, 21, 27, 31, 37, 42, 48, 52, 60, 64, 70, 76, 83, 87, 95, 99, 107, 113, 119, 123, 133, 138, 144, 150, 158, 162, 172, 176, 184, 190, 196, 202, 213, 217, 223, 229, 239, 243, 253, 257, 265, 273, 279, 283, 295, 300, 308, 314, 322, 326, 336, 342, 352

Offset: 0

Author

Ralf Stephan, May 06 2005

Keywords

• nonn
• easy

Comments

Number of ordered triples [k,l,m] with n = k+l*m and k, l, m all in the range [0..n].

From R. J. Mathar, Jun 30 2013: (Start)

A010766 is the following array A read by antidiagonals:

1, 1, 1, 1, 1, 1, ...

2, 1, 1, 1, 1, 1, ...

3, 2, 1, 1, 1, 1, ...

4, 2, 2, 1, 1, 1, ...

5, 3, 2, 2, 1, 1, ...

6, 3, 2, 2, 2, 1, ...

and apparently a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)

Examples

			0+1*2 = 0+2*1 = 1+1*1 = 2+0*0 = 2+0*1 = 2+0*2 = 2+1*0 = 2+2*0 = 2, so a(2)=8.
a(3)=12: 3+0*0, 3+0*m (6), 2+1*1, 1+2*1 (2), 0+3*1 (2).
		

Links

• R. J. Mathar, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A006218, A106634, A106846, A106847.

Cf. A005408, A006218.

Programs

Maple

A106633 := proc(n)
    local a, k, l, m ;
    a := 0 ;
    for k from 0 to n do
        for l from 0 to n do
            if l = 0 then
                if k = n then
                    a := a+n+1 ;
                end if;
            else
                m := (n-k)/l ;
                if type(m,'integer') then
                    a := a+1 ;
                end if;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

Mathematica

A106633[n_] := Module[{a, m}, a = 0; Do[If[l == 0, If[k == n, a = a + n + 1], m = (n - k)/l; If[IntegerQ[m], a = a + 1]], {k, 0, n}, {l, 0, n}]; a];
Table[A106633[n], {n, 0, 56}] (* Jean-François Alcover, Jun 10 2023, after R. J. Mathar *)

PARI

list(n)={
    my(v=vector(n),t);
    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,k,2*k+1+t+=v[k]))
}; \\ Charles R Greathouse IV, Oct 17 2012

Formula

From Ridouane Oudra, Apr 22 2024: (Start)

a(n) = 2*n + 1 + Sum_{k=1..n} floor(n/k);

a(n) = 2*n + 1 + Sum_{k=1..n} tau(k);

a(n) = A005408(n) + A006218(n). (End)

Extensions

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