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

1,1

Comments

The nonnegative integers not occurring in A001045.

A147612(a(n)) = 0.

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

Links

Table of n, a(n) for n=1..71.

Wikipedia, Lambek-Moser theorem

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 6.

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

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 *)

Prog

(Python)
def A147613(n): return (m:=n-2+(k:=(3*n+1).bit_length()))+(m>=((1<<k)|1)//3) # Chai Wah Wu, Apr 17 2025
(PARI) a(n)= my(k=logint(3*n+1, 2), m=n-1+k); m+(m>=(2<<k+1)\3); \\ Ruud H.G. van Tol, Mar 18 2026; after Python

Crossrefs

Cf. A001045.

Sequence in context: A299491 A039098 A015860 * A155939 A114145 A153351

Adjacent sequences: A147610 A147611 A147612 * A147614 A147615 A147616

Keyword

nonn

Author

Reinhard Zumkeller, Nov 08 2008

Status

approved