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A060432
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Partial sums of A002024.
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14
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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In other words, first differences give A002024.
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LINKS
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FORMULA
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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
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
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
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EXAMPLE
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a(7) = 1 + 2 + 2 + 3 + 3 + 3 + 4 = 18.
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MAPLE
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ListTools:-PartialSums([seq(n$n, n=1..10)]); # Robert Israel, Jan 28 2016
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MATHEMATICA
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PROG
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(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
(Haskell)
a060432 n = sum $ zipWith (*) [n, n-1..1] a010054_list
(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Robert A. Stump (bobess(AT)netzero.net), Apr 06 2001
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EXTENSIONS
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STATUS
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approved
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