

A003059


k appears 2k1 times. Also, square root of n, rounded up.


49



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 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
Michael 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_{k=0..n1} A010052(k).  Reinhard Zumkeller, Mar 01 2009
Sum_{n>=1} (1)^(n+1)/a(n) = log(2) (A002162).  Amiram Eldar, Sep 29 2022


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 *)
Ceiling[Sqrt[Range[100]]] (* G. C. Greubel, Nov 14 2018 *)
Table[PadRight[{}, 2k1, k], {k, 10}]//Flatten (* Harvey P. Dale, Jun 07 2020 *)


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
(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, A002162, A002522, A010052, A157466.
Sequence in context: A083375 A088519 A135034 * A325678 A247189 A192002
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



