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A003059
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k appears 2k-1 times. Also, square root of n, rounded up.
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50
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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)
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = ceiling(sqrt(n)).
G.f.: (Sum_{n>=0} x^(n^2)) * x/(1-x). - Michael Somos, May 03 2003
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MAPLE
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MATHEMATICA
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Table[PadRight[{}, 2k-1, k], {k, 10}]//Flatten (* Harvey P. Dale, Jun 07 2020 *)
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PROG
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(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..]
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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