|
%I #70 Oct 17 2022 01:43:41
%S 1,3,5,8,11,14,18,22,26,30,35,40,45,50,55,61,67,73,79,85,91,98,105,
%T 112,119,126,133,140,148,156,164,172,180,188,196,204,213,222,231,240,
%U 249,258,267,276,285,295,305,315,325,335,345,355,365,375,385,396,407,418
%N Partial sums of A002024.
%C In other words, first differences give A002024.
%C Equals A010054 convolved with [1, 2, 3, ...]. - _Gary W. Adamson_, Mar 16 2010
%H Harry J. Smith, <a href="/A060432/b060432.txt">Table of n, a(n) for n=1..1000</a>
%H Gorka Zamora-López and Romain Brasselet, <a href="https://arxiv.org/1810.12825">Sizing the length of complex networks</a>, arXiv:1810.12825 [physics.soc-ph], 2018.
%F Let f(n) = floor(1/2 + sqrt(2*n)), then this function is S(n) = f(1) + f(2) + f(3) + ... + f(n).
%F a(n) is asymptotic to c*n^(3/2) with c=0.9428.... - _Benoit Cloitre_, Dec 18 2002
%F a(n) is asymptotic to c*n^(3/2) with c = (2/3)*sqrt(2) = .942809.... - _Franklin T. Adams-Watters_, Sep 07 2006
%F Set R = round(sqrt(2*n)), then a(n) = ((6*n+1)*R-R^3)/6. - _Gerald Hillier_, Nov 28 2008
%F 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
%F 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
%F 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
%F G.f.: (x/(1 - x)^2)*Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, May 30 2017
%F 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
%e a(7) = 1 + 2 + 2 + 3 + 3 + 3 + 4 = 18.
%p ListTools:-PartialSums([seq(n$n,n=1..10)]); # _Robert Israel_, Jan 28 2016
%t a[n_] := Sum[Floor[1/2 + Sqrt[2*k]], {k, 1, n}]; Array[a, 60] (* _Jean-François Alcover_, Jan 10 2016 *)
%o (PARI) f(n) = floor(1/2+sqrt(2*n))
%o for(n=1,100,print1(sum(k=1,n,f(k)),","))
%o (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
%o (Haskell)
%o a060432 n = sum $ zipWith (*) [n,n-1..1] a010054_list
%o -- _Reinhard Zumkeller_, Dec 17 2011
%o (Python)
%o from math import isqrt
%o 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
%Y Cf. A002024, A006463, A010054.
%K easy,nonn
%O 1,2
%A Robert A. Stump (bobess(AT)netzero.net), Apr 06 2001
%E More terms from _Jason Earls_, Jan 08 2002
|
