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A003059 n appears 2n-1 times. 27
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 (list; graph; refs; listen; history; text; internal format)
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 Erdős-Szekeres theorem. - Lekraj Beedassy, May 20 2003

With offset 0, apparently the least k such that binomial(2n,n-k) < (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/(1-x). - 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 (* Jean-François Alcover, Jun 10 2013 *)

PROG

(PARI) a(n)=if(n<1, 0, 1+sqrtint(n-1))

(Haskell)

a003059 n = a003059_list !! (n-1)

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

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Last modified September 26 06:54 EDT 2017. Contains 292502 sequences.