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

1, 3, 6, 7, 11, 13, 18, 21, 22, 28, 32, 34, 41, 46, 49, 50, 58, 64, 68, 70, 79, 86, 91, 94, 95, 105, 113, 119, 123, 125, 136, 145, 152, 157, 160, 161, 173, 183, 191, 197, 201, 203, 216, 227, 236, 243, 248, 251, 252, 266, 278, 288, 296, 302, 306, 308, 323, 336

Offset

1,2

Comments

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

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

Cumulative sums of A122196. - Franklin T. Adams-Watters, Aug 25 2006

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

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

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

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

Mathematica

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)]]

Prog

(PARI) a(n)=if(n<2, n>0, n+a(n-sqrtint(n)))
(Haskell)
a076644 n = a076644_list !! (n-1)
a076644_list = scanl1 (+) a122196_list

Crossrefs

Cf. A000196, A076660, A122196.

Sequence in context: A153033 A353240 A282167 * A214961 A189626 A338066

Adjacent sequences: A076641 A076642 A076643 * A076645 A076646 A076647

Keyword

nonn

Author

Benoit Cloitre, Oct 23 2002

Status

approved