A003059 k appears 2k-1 times. Also, square root of n, rounded up.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Also the burning number of the cycle graph C_n (for n >= 4) and the path graph (for n >= 1). - Eric W. Weisstein, Jan 10 2024

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Michael Somos, Sequences used for indexing triangular or square arrays.
• Eric Weisstein's World of Mathematics, Burning Number
• Eric Weisstein's World of Mathematics, Cycle Graph
• Eric Weisstein's World of Mathematics, Path Graph

Crossrefs

Cf. A000196, A000290, A002162, A002522, A010052, A157466.

Programs

Haskell

a003059 n = a003059_list !! (n-1)
a003059_list = concat $ zipWith ($) (map replicate [1,3..]) [1..]
-- Reinhard Zumkeller, Mar 18 2011

Magma

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

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 *)
Table[PadRight[{},2k-1,k],{k,10}]//Flatten (* Harvey P. Dale, Jun 07 2020 *)

PARI

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

Python

from math import isqrt
def A003059(n): return isqrt(n-1)+1 # Chai Wah Wu, Nov 14 2022

Sage

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

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

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Sep 29 2022

Extensions

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