|
%I #37 Oct 01 2022 06:15:18
%S 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,
%T 11,11,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,13,15,15,15,15,15,
%U 15,15,15,15,15,15,15,15,15,15,17,17,17,17,17,17,17,17,17,17,17,17,17
%N k appears k times (k odd).
%C 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
%C The burning number of a connected graph of order n is at most a(n). See Bessy et al. - _Michel Marcus_, Jun 18 2018
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
%H T. D. Noe, <a href="/A001650/b001650.txt">Table of n, a(n) for n = 1..10000</a>
%H Stéphane Bessy, Anthony Bonato, Jeannette Janssen and Dieter Rautenbach, <a href="https://arxiv.org/abs/1511.06023">Bounds on the Burning Number</a>, arXiv:1511.06023 [math.CO], 2015-2016.
%H Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, <a href="https://arxiv.org/abs/1202.0276">A combinatorial approach for solving certain nested recursions with non-slow solutions</a>, arXiv preprint arXiv:1202.0276 [math.CO], 2012.
%F a(n) = 1 + 2*floor(sqrt(n-1)), n > 0. - _Antonio Esposito_, Jan 21 2002
%F From _Michael Somos_, Apr 29 2003: (Start)
%F G.f.: theta_3(x)*x/(1-x).
%F a(n+1) = a(n) + A000122(n). (End)
%F a(1) = 1, a(2) = 3, a(3) = 3, a(n) = a(n-a(n-2))+2. - _Branko Curgus_, May 07 2010
%F a(n) = 2*ceiling(sqrt(n)) - 1. - _Branko Curgus_, May 07 2010
%F Seen as a triangle read by rows: T(n,k) = 2*(n-1), k=1..n. - _Reinhard Zumkeller_, Nov 14 2015
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - _Amiram Eldar_, Oct 01 2022
%t a[1]=1,a[2]=3,a[3]=3,a[n_]:=a[n]=a[n-a[n-2]]+2 (* _Branko Curgus_, May 07 2010 *)
%t Flatten[Table[Table[n,{n}],{n,1,17,2}]] (* _Harvey P. Dale_, Mar 31 2013 *)
%o (PARI) a(n)=if(n<1,0,1+2*sqrtint(n-1))
%o (Haskell)
%o a001650 n k = a001650_tabf !! (n-1) !! (k-1)
%o a001650_row n = a001650_tabf !! (n-1)
%o a001650_tabf = iterate (\xs@(x:_) -> map (+ 2) (x:x:xs)) [1]
%o a001650_list = concat a001650_tabf
%o -- _Reinhard Zumkeller_, Nov 14 2015
%Y Partial sums of A000122.
%Y Cf. A001670, A003881, A111650, A131507, A193832.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_, Apr 29 2003
|
