A060432 Partial sums of A002024.

1, 3, 5, 8, 11, 14, 18, 22, 26, 30, 35, 40, 45, 50, 55, 61, 67, 73, 79, 85, 91, 98, 105, 112, 119, 126, 133, 140, 148, 156, 164, 172, 180, 188, 196, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 295, 305, 315, 325, 335, 345, 355, 365, 375, 385, 396, 407, 418

Offset: 1

Author

Robert A. Stump (bobess(AT)netzero.net), Apr 06 2001

Keywords

• easy
• nonn

Comments

In other words, first differences give A002024.

Equals A010054 convolved with [1, 2, 3, ...]. - Gary W. Adamson, Mar 16 2010

Examples

			a(7) = 1 + 2 + 2 + 3 + 3 + 3 + 4 = 18.
		

Links

• Harry J. Smith, Table of n, a(n) for n=1..1000
• Gorka Zamora-López and Romain Brasselet, Sizing the length of complex networks, arXiv:1810.12825 [physics.soc-ph], 2018.

Crossrefs

Cf. A002024, A006463, A010054.

Programs

Haskell

a060432 n = sum $ zipWith (*) [n,n-1..1] a010054_list
-- Reinhard Zumkeller, Dec 17 2011

Maple

ListTools:-PartialSums([seq(n$n,n=1..10)]); # Robert Israel, Jan 28 2016

Mathematica

a[n_] := Sum[Floor[1/2 + Sqrt[2*k]], {k, 1, n}]; Array[a, 60] (* Jean-François Alcover, Jan 10 2016 *)
Accumulate[Table[PadRight[{},n,n],{n,15}]//Flatten] (* Harvey P. Dale, May 24 2025 *)

PARI

f(n) = floor(1/2+sqrt(2*n))
for(n=1,100,print1(sum(k=1,n,f(k)),","))

PARI

{ default(realprecision, 100); for (n=1, 1000, a=sum(k=1, n, floor(1/2 + sqrt(2*k))); write("b060432.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009

Python

from math import isqrt
def A060432(n): return (k:=(r:=isqrt(m:=n+1<<1))+int((m<<2)>(r<<2)*(r+1)+1)-1)*(k*(-k - 3) + 6*n - 2)//6 + n # Chai Wah Wu, Oct 16 2022

Formula

Let f(n) = floor(1/2 + sqrt(2*n)), then this function is S(n) = f(1) + f(2) + f(3) + ... + f(n).

a(n) is asymptotic to c*n^(3/2) with c=0.9428.... - Benoit Cloitre, Dec 18 2002

a(n) is asymptotic to c*n^(3/2) with c = (2/3)*sqrt(2) = .942809.... - Franklin T. Adams-Watters, Sep 07 2006

Set R = round(sqrt(2*n)), then a(n) = ((6*n+1)*R-R^3)/6. - Gerald Hillier, Nov 28 2008

G.f.: W(0)/(2*(1-x)^2), where W(k) = 1 + 1/( 1 - x^(k+1)/( x^(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013

a(n) = A000330(A003056(n)) + (A003056(n) + 1) * (n - A057944(n)). This represents a closed form, because all of the constituent sequences (i.e., A003056, A000330, A057944) have a known closed form. - Peter Kagey, Jan 28 2016

G.f.: x^(7/8)*Theta_2(0,x^(1/2))/(2*(1-x)^2) where Theta_2 is a Jacobi theta function. - Robert Israel, Jan 28 2016

G.f.: (x/(1 - x)^2)*Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, May 30 2017

a(n) = n*(k+1)-k*(k+1)*(k+2)/6 where k = A003056(n) is the largest integer such that k*(k+1)/2 <= n. - Bogdan Blaga, Feb 04 2021

Extensions

More terms from Jason Earls, Jan 08 2002