%I #42 Aug 04 2024 17:02:13

%S 1,5,12,26,42,73,102,152,204,278,345,464,556,693,835,1021,1175,1422,

%T 1613,1907,2173,2496,2773,3228,3569,4015,4445,4998,5434,6120,6617,

%U 7331,7965,8717,9391,10392,11096,12031,12909,14059,14921,16219,17166,18489,19711,21072,22201

%N a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

%F a(n) = Sum_{k=1..n} binomial(k+1,2) * floor(n/k).

%F G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^3 = 1/(1-x) * Sum_{k>=1} binomial(k+1,2) * x^k/(1-x^k).

%F a(n) = (A064602(n)+A024916(n))/2. - _Chai Wah Wu_, Oct 26 2023

%t Table[Sum[Binomial[Floor[n/k+2],3],{k,n}],{n,50}] (* _Harvey P. Dale_, Aug 04 2024 *)

%o (PARI) a(n) = sum(k=1, n, binomial(n\k+2, 3));

%o (Python)

%o from math import isqrt

%o def A364970(n): return (-(s:=isqrt(n))**2*(s+1)*(s+2)+sum((q:=n//k)*(3*k*(k+1)+(q+1)*(q+2)) for k in range(1,s+1)))//6 # _Chai Wah Wu_, Oct 26 2023

%Y Partial sums of A007437.

%Y Cf. A006218, A024916, A064602, A365409, A365439.

%K nonn,easy

%O 1,2

%A _Seiichi Manyama_, Oct 23 2023