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A076644 a(1)=1; for n>1, a(n) = a(n-floor(sqrt(n))) + n.

%I A076644 #30 May 17 2025 19:22:05
%S A076644 1,3,6,7,11,13,18,21,22,28,32,34,41,46,49,50,58,64,68,70,79,86,91,94,
%T A076644 95,105,113,119,123,125,136,145,152,157,160,161,173,183,191,197,201,
%U A076644 203,216,227,236,243,248,251,252,266,278,288,296,302,306,308,323,336
%N A076644 a(1)=1; for n>1, a(n) = a(n-floor(sqrt(n))) + n.
%C A076644 a(n) = floor(2/3*n*(sqrt(n)+1)) for n in A076660.
%C A076644 The sign of a(n) - floor(2/3*n*(sqrt(n)+1)) changes often.
%C A076644 Cumulative sums of A122196. - _Franklin T. Adams-Watters_, Aug 25 2006
%H A076644 Reinhard Zumkeller, <a href="/A076644/b076644.txt">Table of n, a(n) for n = 1..10000</a>
%F A076644 Write n=r^2+s with -r < s <= r; then a(n) = r*(r+1)*(4r-1)/6 + x, where x = -s^2 if s <= 0, x = s*(2r+1-s) if s >= 0. - _Dean Hickerson_, Nov 11 2002
%F A076644 a(n) is asymptotic to 2/3*n^(3/2).
%F A076644 a(n) = n*(2*x^2+2*x+1-n) - 1/6*x*(x+1)*(6*x^2+2*x+1) + floor((n-x^2)/(x+1))*(2*x+1)*(n-x-x^2) where x = floor(sqrt(n)). - _Hoang Xuan Thanh_, May 17 2025
%t A076644 a[n_] := Module[{r, s}, r=Floor[1/2+Sqrt[n]]; s=n-r^2; (r(r+1)(4r-1))/6+If[s<=0, -s^2, s(2r+1-s)]]
%o A076644 (PARI) a(n)=if(n<2,n>0,n+a(n-sqrtint(n)))
%o A076644 (Haskell)
%o A076644 a076644 n = a076644_list !! (n-1)
%o A076644 a076644_list = scanl1 (+) a122196_list
%Y A076644 Cf. A000196, A076660, A122196.
%K A076644 nonn
%O A076644 1,2
%A A076644 _Benoit Cloitre_, Oct 23 2002