cp's OEIS Frontend

A106846 a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.

%I A106846 #23 Aug 03 2024 19:20:14
%S A106846 0,4,22,64,144,269,461,720,1072,1522,2092,2774,3626,4614,5776,7126,
%T A106846 8694,10445,12461,14684,17204,19997,23077,26412,30156,34206,38600,
%U A106846 43352,48532,54042,60072,66458,73338,80664,88450,96710,105638,114999
%N A106846 a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.
%H A106846 R. J. Mathar, <a href="/A106846/b106846.txt">Table of n, a(n) for n = 0..1000</a>
%F A106846 From _Ridouane Oudra_, Jun 24 2024: (Start)
%F A106846 a(n) = (1/2) * (n*(n+1)*(2*n+1) + Sum_{k=1..n} (n^2 + n + k - k^2) * tau(k)).
%F A106846 a(n) = (1/2) * (A055112(n) + (n^2 + n) * A006218(n) + A143127(n) - A319085(n)).
%F A106846 a(n) = A059270(n) + A143127(n) + A106847(n). (End)
%p A106846 A106846 := proc(n)
%p A106846     local a,k,l,m ;
%p A106846     a := 0 ;
%p A106846     for k from 0 to n do
%p A106846         for l from 0 to n do
%p A106846             if l = 0 then
%p A106846                 m := n ;
%p A106846             else
%p A106846                 m := floor((n-k)/l) ;
%p A106846             end if;
%p A106846             if m >=0 then
%p A106846                 m := min(m,n) ;
%p A106846                 a := a+(m+1)*k+l*m*(m+1)/2 ;
%p A106846             end if;
%p A106846         end do:
%p A106846     end do:
%p A106846     a ;
%p A106846 end proc: # _R. J. Mathar_, Oct 17 2012
%t A106846 A106846[n_] := Module[{a, k, l, m }, a = 0; For[k = 0, k <= n, k++, For[l = 0, l <= n, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 0, m = Min[m, n]; a = a + (m + 1)*k + l*m*(m + 1)/2 ]]]; a];
%t A106846 Table[A106846[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 04 2024, after _R. J. Mathar_ *)
%Y A106846 Cf. A106633, A106634, A106847.
%Y A106846 Cf. A055112, A006218, A143127, A319085, A059270, A143127.
%K A106846 nonn
%O A106846 0,2
%A A106846 _Ralf Stephan_, May 06 2005