cp's OEIS Frontend

A080344 Partial sums of A023969.

%I A080344 #27 Jun 09 2025 00:26:42
%S A080344 0,0,0,1,1,1,1,2,3,3,3,3,3,4,5,6,6,6,6,6,6,7,8,9,10,10,10,10,10,10,10,
%T A080344 11,12,13,14,15,15,15,15,15,15,15,15,16,17,18,19,20,21,21,21,21,21,21,
%U A080344 21,21,21,22,23,24,25,26,27,28,28,28,28,28,28,28,28,28,28,29,30,31,32,33,34,35
%N A080344 Partial sums of A023969.
%F A080344 From _Ridouane Oudra_, May 11 2019: (Start)
%F A080344 a(n) = (1/2)*(n + 1 - t - abs(n + 1 - t^2)), where t = floor(sqrt(n+1) + 1/2).
%F A080344 a(n) = (1/2)*(n + 1 - A000194(n+1) - abs(n + 1 - A000194(n+1)^2)).
%F A080344 a(n) = (1/2)*(A056847(n+1) - A053188(n+1)). (End)
%o A080344 (Magma) [1/2*(n+1-Floor(Sqrt(n+1)+1/2)-Abs(n+1-(Floor(Sqrt(n+1)+1/2))^2)):n in [0..90]]; // _Marius A. Burtea_, May 09 2019
%o A080344 (PARI) f(n) = sqrtint(4*n)-2*sqrtint(n); \\ A023969
%o A080344 a(n) = sum(k=0, n, f(k)); \\ _Michel Marcus_, May 10 2019
%o A080344 (Python)
%o A080344 from math import isqrt
%o A080344 def A080344(n): return n+1-(k:=(m:=isqrt(n+1))+int(n>=m*(m+1)))-abs(n+1-k**2)>>1 # _Chai Wah Wu_, Jun 05 2025
%Y A080344 Cf. A000194, A023969, A053188, A056847.
%K A080344 nonn
%O A080344 0,8
%A A080344 _N. J. A. Sloane_, Mar 20 2003