A130829 2n+1 appears 2n times.

3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 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, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17

Offset: 1

Author

Paul Curtz, Jul 17 2007

Keywords

• nonn
• easy

Crossrefs

Cf. A001670, A002024, A003057.

Programs

Maple

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

Python

from math import isqrt
def A130829(n): return 1|((m:=isqrt(n))+int((n-m*(m+1)<<2)>=1)<<1) # Chai Wah Wu, Oct 17 2022

Formula

a(n) = 2*floor(sqrt(n)+1/2)+1. - Mikael Aaltonen, Jan 14 2015

From Robert Israel, Jan 14 2015: (Start)

G.f.: (x/(1-x))*(1+2*Sum_{m>=0} x^(m*(m+1))) = (x/(1-x))*(1+x^(-1/4)*theta_2(0,x)) where theta_2 is the second Jacobi theta function.

a(2n) = a(2n-1) = 2*A002024(n)+1.

a(n) = A001670(n)+1.

(End)