A001670 k appears k times (k even).

2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Formula

a(n) = 2*floor(1/2 + sqrt(n)). - Antonio Esposito, Jan 21 2002; corrected by Branko Curgus, May 11 2010

With a different offset: g.f. = Sum_{j>=0} 2*x^(j^2+i)/(1-x). - Ralf Stephan, Mar 11 2003

From Branko Curgus, May 11 2010: (Start)

a(n) = a(n - a(n-2)) + 2; a(1)=2, a(2)=2.

a(n) = 2*round(sqrt(n)). (End)

G.f.: x^(3/4)*theta_2(0,x)/(1-x) where theta_2 is the second Jacobi theta function. - Robert Israel, Jan 14 2015

a(n) = 2*floor((sqrt(4*n-3)+1)/2). - Néstor Jofré, Apr 24 2017

Maple

seq(2*n $ 2*n, n = 1 .. 10); # Robert Israel, Jan 14 2015

Mathematica

a[1]=2, a[2]=2, a[n_]:=a[n]=a[n-a[n-2]]+2 (* Branko Curgus, May 11 2010 *)
Flatten[Table[Table[n, {n}], {n, 2, 16, 2}]] (* Harvey P. Dale, May 31 2012 *)

Prog

(Magma) [2*Round(Sqrt(n)): n in [1..70]]; // Vincenzo Librandi, Jun 23 2011
(PARI) a(n)=round(sqrt(n))<<1 \\ Charles R Greathouse IV, Jun 23 2011
(MATLAB) a = @(n) 2*floor((sqrt(4*n-3)+1)/2); % handle function // Néstor Jofré, Apr 24 2017
(Python)
from math import isqrt
def A001670(n): return (m:=isqrt(n))+int((n-m*(m+1)<<2)>=1)<<1 # Chai Wah Wu, Jul 29 2022

Crossrefs

Cf. A001650, A000194.

Equals A130829(n) - 1.

Sequence in context: A035683 A261009 A239896 * A100144 A076222 A177692

Adjacent sequences: A001667 A001668 A001669 * A001671 A001672 A001673

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Offset changed from 2 to 1 by Vincenzo Librandi, Jun 23 2011

Status

approved