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A003059 k appears 2k-1 times. Also, square root of n, rounded up. 35
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_{k=0..n-1} A010052(k). - 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 *)

Ceiling[Sqrt[Range[100]]] (* G. C. Greubel, Nov 14 2018 *)

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

(Sage) [ceil(sqrt(n)) for n in (1..100)] # G. C. Greubel, Nov 14 2018

(MAGMA) [Ceiling(Sqrt(n)): n in [1..100]]; // G. C. Greubel, Nov 14 2018

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,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name edited by M. F. Hasler, Nov 13 2018

STATUS

approved

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Last modified November 15 04:00 EST 2018. Contains 317225 sequences. (Running on oeis4.)