A001650 - OEIS (original) (raw)

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

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

Stéphane Bessy, Anthony Bonato, Jeannette Janssen and Dieter Rautenbach, Bounds on the Burning Number, arXiv:1511.06023 [math.CO], 2015-2016.

FORMULA

a(n) = 1 + 2*floor(sqrt(n-1)), n > 0. - Antonio Esposito, Jan 21 2002

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

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

(Python)

from math import isqrt