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A001650
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k appears k times (k odd).
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18
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1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17
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OFFSET
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1,2
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COMMENTS
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For n >= 0, a(n+1) is the number of integers x with |x| <= sqrt(n), or equivalently the number of points in the Z^1 lattice of norm <= n+1. - David W. Wilson, Oct 22 2006
The burning number of a connected graph of order n is at most a(n). See Bessy et al. - Michel Marcus, Jun 18 2018
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Stéphane Bessy, Anthony Bonato, Jeannette Janssen and Dieter Rautenbach, Bounds on the Burning Number, arXiv:1511.06023 [math.CO], 2015-2016.
Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, A combinatorial approach for solving certain nested recursions with non-slow solutions, arXiv preprint arXiv:1202.0276 [math.CO], 2012.
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FORMULA
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a(n) = 1 + 2*floor(sqrt(n-1)), n > 0. - Antonio Esposito, Jan 21 2002
From Michael Somos, Apr 29 2003: (Start)
G.f.: theta_3(x)*x/(1-x).
a(n+1) = a(n) + A000122(n). (End)
a(1) = 1, a(2) = 3, a(3) = 3, a(n) = a(n-a(n-2))+2. - Branko Curgus, May 07 2010
a(n) = 2*ceiling(sqrt(n)) - 1. - Branko Curgus, May 07 2010
Seen as a triangle read by rows: T(n,k) = 2*(n-1), k=1..n. - Reinhard Zumkeller, Nov 14 2015
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - Amiram Eldar, Oct 01 2022
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MATHEMATICA
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a[1]=1, a[2]=3, a[3]=3, a[n_]:=a[n]=a[n-a[n-2]]+2 (* Branko Curgus, May 07 2010 *)
Flatten[Table[Table[n, {n}], {n, 1, 17, 2}]] (* Harvey P. Dale, Mar 31 2013 *)
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PROG
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(PARI) a(n)=if(n<1, 0, 1+2*sqrtint(n-1))
(Haskell)
a001650 n k = a001650_tabf !! (n-1) !! (k-1)
a001650_row n = a001650_tabf !! (n-1)
a001650_tabf = iterate (\xs@(x:_) -> map (+ 2) (x:x:xs)) [1]
a001650_list = concat a001650_tabf
-- Reinhard Zumkeller, Nov 14 2015
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CROSSREFS
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Partial sums of A000122.
Cf. A001670, A003881, A111650, A131507, A193832.
Sequence in context: A136800 A126661 A162226 * A130175 A200266 A101290
Adjacent sequences: A001647 A001648 A001649 * A001651 A001652 A001653
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Michael Somos, Apr 29 2003
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STATUS
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approved
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