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 A003059 k appears 2k-1 times. Also, square root of n, rounded up. 48
 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 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 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/(1-x). - Michael Somos, May 03 2003 a(n) = Sum_{k=0..n-1} A010052(k). - Reinhard Zumkeller, Mar 01 2009 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 *) 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 CROSSREFS Cf. A000196, A000290, A157466. Sequence in context: A083375 A088519 A135034 * A325678 A247189 A192002 Adjacent sequences:  A003056 A003057 A003058 * A003060 A003061 A003062 KEYWORD nonn,easy AUTHOR EXTENSIONS Name edited by M. F. Hasler, Nov 13 2018 STATUS approved

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Last modified May 13 04:09 EDT 2021. Contains 343836 sequences. (Running on oeis4.)