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
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
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
MAPLE
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 *)
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
(Python)
from math import isqrt
def A003059(n): return isqrt(n-1)+1 # Chai Wah Wu, Nov 14 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Name edited by M. F. Hasler, Nov 13 2018
STATUS
approved