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a(1)=1; for n>1, a(n) = a(n-floor(sqrt(n))) + n.
3

%I #22 Feb 16 2021 09:29:58

%S 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 95,105,113,119,123,125,136,145,152,157,160,161,173,183,191,197,201,

%U 203,216,227,236,243,248,251,252,266,278,288,296,302,306,308,323,336

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

%C a(n) = floor(2/3*n*(sqrt(n)+1)) for n in A076660.

%C The sign of a(n) - floor(2/3*n*(sqrt(n)+1)) changes often.

%C Cumulative sums of A122196. - _Franklin T. Adams-Watters_, Aug 25 2006

%H Reinhard Zumkeller, <a href="/A076644/b076644.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 a(n) is asymptotic to 2/3*n^(3/2).

%t 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 (PARI) a(n)=if(n<2,n>0,n+a(n-sqrtint(n)))

%o (Haskell)

%o a076644 n = a076644_list !! (n-1)

%o a076644_list = scanl1 (+) a122196_list

%Y Cf. A000196, A076660, A122196.

%K nonn

%O 1,2

%A _Benoit Cloitre_, Oct 23 2002