

A003059


n appears 2n1 times.


33



1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10
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OFFSET

1,2


COMMENTS

n+1 first appears in the sequence at the A002522(n)th entry (since the ultimate occurrence of n is n^2). a(n) refers to the greatest minimal length of monotone subsequence (i.e.either increasing or decreasing) contained within any sequence of n distinct numbers,according to the ErdősSzekeres theorem.  Lekraj Beedassy, May 20 2003
With offset 0, apparently the least k such that binomial(2n,nk) < (1/e) binomial(2n,n).  T. D. Noe, Mar 12 2009
a(n) is the number of nonnegative integer solutions of equation x + y^2 = n  1.  Ran Pan, Oct 02 2015


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
M. Somos, Sequences used for indexing triangular or square arrays


FORMULA

a(n) = ceiling(sqrt(n)).
G.f.: (Sum_{n>=0} x^(n^2))x/(1x).  Michael Somos, May 03, 2003
a(n) = SUM(A010052(k): 0 <= k < n).  Reinhard Zumkeller, Mar 01 2009


MAPLE

A003059:=n>ceil(sqrt(n)); seq(A003059(k), k=1..100); # Wesley Ivan Hurt, Nov 08 2013


MATHEMATICA

Table[ Table[n, {2n  1}], {n, 1, 10}] // Flatten (* JeanFrançois Alcover, Jun 10 2013 *)


PROG

(PARI) a(n)=if(n<1, 0, 1+sqrtint(n1))
(Haskell)
a003059 n = a003059_list !! (n1)
a003059_list = concat $ zipWith ($) (map replicate [1, 3..]) [1..]
 Reinhard Zumkeller, Mar 18 2011


CROSSREFS

Cf. A000196, A000290, A157466.
Sequence in context: A083375 A088519 A135034 * A247189 A192002 A263270
Adjacent sequences: A003056 A003057 A003058 * A003060 A003061 A003062


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



