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