A165453 Linear interpolation of the sequence that maps an entry of A002378 to the corresponding entry of A006331.

0, 1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 288, 296, 304

Offset

0,3

Links

Table of n, a(n) for n=0..59.

Formula

a(0)=0, a(n) = max(min(a(n-1)+x, n+a(n-x))), where the maximum is taken over all values for x from 1 to n.

a(n) = Sum_{k=1..n} floor(sqrt(k)+1/2). - Wesley Ivan Hurt, Dec 01 2020

a(n) = (1/3)*t*(3*n + 1 - t^2), where t = floor(sqrt(n)+1/2). - Ridouane Oudra, Feb 22 2021

Mathematica

f[n_]:=Round[Sqrt[n]]; a=0; lst={}; Do[AppendTo[lst, a+=f[n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
Table[Sum[Floor[Sqrt[n + 1 - k] + 1/2], {k, n + 1}], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 01 2020 *)

Prog

(Python)
from math import isqrt
def A165453(n): return (k:=(m:=isqrt(n))+(n-m*(m+1)>=1))*(3*n+1-k**2)//3 # Chai Wah Wu, Jun 19 2024

Crossrefs

Cf. A002378, A006331.

Partial sums of A000194.

Sequence in context: A130174 A061168 A130798 * A282168 A025224 A294023

Adjacent sequences: A165450 A165451 A165452 * A165454 A165455 A165456

Keyword

easy,nonn

Author

Friedrich Regen (friedrich.regen(AT)tu-ilmenau.de), Sep 20 2009

Status

approved