A001650 k appears k times (k odd).

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

Offset

1,2

Comments

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

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.

Links

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.

Formula

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

Mathematica

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 *)

Prog

(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

Crossrefs

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

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos, Apr 29 2003

Status

approved