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%I #75 Jan 10 2024 09:10:02
%S 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,
%T 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,
%U 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
%N k appears 2k-1 times. Also, square root of n, rounded up.
%C 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
%C With offset 0, apparently the least k such that binomial(2n,n-k) < (1/e) binomial(2n,n). - _T. D. Noe_, Mar 12 2009
%C a(n) is the number of nonnegative integer solutions of equation x + y^2 = n - 1. - _Ran Pan_, Oct 02 2015
%C 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
%H T. D. Noe, <a href="/A003059/b003059.txt">Table of n, a(n) for n = 1..1000</a>
%H Michael Somos, <a href="/A073189/a073189.txt">Sequences used for indexing triangular or square arrays</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BurningNumber.html">Burning Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%F a(n) = ceiling(sqrt(n)).
%F G.f.: (Sum_{n>=0} x^(n^2)) * x/(1-x). - _Michael Somos_, May 03 2003
%F a(n) = Sum_{k=0..n-1} A010052(k). - _Reinhard Zumkeller_, Mar 01 2009
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - _Amiram Eldar_, Sep 29 2022
%p A003059:=n->ceil(sqrt(n)); seq(A003059(k), k=1..100); # _Wesley Ivan Hurt_, Nov 08 2013
%t Table[ Table[n, {2n - 1}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Jun 10 2013 *)
%t Ceiling[Sqrt[Range[100]]] (* _G. C. Greubel_, Nov 14 2018 *)
%t Table[PadRight[{},2k-1,k],{k,10}]//Flatten (* _Harvey P. Dale_, Jun 07 2020 *)
%o (PARI) a(n)=if(n<1,0,1+sqrtint(n-1))
%o (Haskell)
%o a003059 n = a003059_list !! (n-1)
%o a003059_list = concat $ zipWith ($) (map replicate [1,3..]) [1..]
%o -- _Reinhard Zumkeller_, Mar 18 2011
%o (Sage) [ceil(sqrt(n)) for n in (1..100)] # _G. C. Greubel_, Nov 14 2018
%o (Magma) [Ceiling(Sqrt(n)): n in [1..100]]; // _G. C. Greubel_, Nov 14 2018
%o (Python)
%o from math import isqrt
%o def A003059(n): return isqrt(n-1)+1 # _Chai Wah Wu_, Nov 14 2022
%Y Cf. A000196, A000290, A002162, A002522, A010052, A157466.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E Name edited by _M. F. Hasler_, Nov 13 2018
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