A147613 Numbers that are not Jacobsthal numbers.

2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Reinhard Zumkeller, Nov 08 2008

Keywords

• nonn

Comments

The nonnegative integers not occurring in A001045.

A147612(a(n)) = 0.

The formula below is a consequence of the Lambek-Moser theorem.

Links

• Wikipedia, Lambek-Moser theorem

Crossrefs

Cf. A001045.

Programs

Maple

a:=proc(n) floor(-LambertW(-1,-1/3*ln(2)*2^(3/2-n))/ln(2)+1/2) end:
seq(a(n), n=1..70); # Simon Plouffe, Nov 29 2017

Mathematica

Complement[Range[m = 100], LinearRecurrence[{1, 2}, {0, 1}, m]] (* Jean-François Alcover, Feb 13 2018 *)

Python

def A147613(n): return (m:=n-2+(k:=(3*n+1).bit_length()))+(m>=((1<Chai Wah Wu, Apr 17 2025

Formula

a(n) = floor(-LambertW(-1, -(1/3)*log(2)*2^(3/2-n))/log(2) + 1/2). - Nicolas Normand (Nicolas.Normand(at)polytech.univ-nantes.fr), Nov 29 2017

a(n) = m+1 if m>=A001045(floor(log_2(3*n+1))+1) and a(n) = m otherwise, where m = n-1+floor(log_2(3*n+1)). - Chai Wah Wu, Apr 17 2025