A118011 Complement of the Connell sequence (A001614); a(n) = 4*n - A001614(n).

3, 6, 8, 11, 13, 15, 18, 20, 22, 24, 27, 29, 31, 33, 35, 38, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 76, 78, 80, 83, 85, 87, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Author

Paul D. Hanna, Apr 10 2006

Keywords

• nonn

Comments

a(n) is the position of the second appearance of n in A117384, where A117384(m) = A117384(k) and k = 4*A117384(m) - m. The Connell sequence (A001614) is generated as: 1 odd, 2 even, 3 odd, ...

Crossrefs

Cf. A001614, A014132, A023531, A117384, A118012.

A171152 gives partial sums.

Programs

Magma

[2*n+Round(Sqrt(2*n)): n in [1..70]]; // Vincenzo Librandi, Apr 16 2015

Mathematica

Table[2 n + Round[Sqrt[2 n]], {n, 70}] (* Vincenzo Librandi, Apr 16 2015 *)

Python

from math import isqrt
def A118011(n): return (m:=n<<1)+(k:=isqrt(m))+int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

Formula

A001614(n) = A118012(a(n)).

a(n) = 2n+[(1+sqrt(8n-7))/2]. - Juri-Stepan Gerasimov Aug 25 2009

a(n) = 2*n+round(sqrt(2*n)). - Gerald Hillier, Apr 16 2015

From Robert Israel, Apr 20 2015 (Start):

a(n) = 2*n + 1 + Sum_{j=0..n-2} A023531(j).

G.f.: 2*x/(1-x)^2 + x/(1-x) * Sum_{j=0..oo} x^(j*(j+1)/2) = 2*x/(1-x)^2 + x^(7/8)/(2-2*x) * Theta2(0,sqrt(x)), where Theta2 is a Jacobi theta function. (End)

a(n) = n+A014132(n). - Chai Wah Wu, Oct 19 2024