A004743 Numbers whose binary expansion does not contain 110.

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 47, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 79, 80, 81, 82, 83, 84, 85, 87, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A003726 (no 111).

Programs

Haskell

a004743 n = a004743_list !! (n-1)
a004743_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 6 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Mathematica

Select[Range[0, 140], !StringContainsQ[IntegerString[#, 2], "110"] &] (* Amiram Eldar, Feb 13 2022 *)
Select[Range[0,150],SequenceCount[IntegerDigits[#,2],{1,1,0}]==0&] (* Harvey P. Dale, Mar 14 2025 *)

PARI

is(n)=n=binary(n);for(i=3,#n,if(!n[i]&&n[i-2]&&n[i-1],return(0))); 1 \\ Charles R Greathouse IV, Mar 26 2013

PARI

is(n)=while(n>5, if(bitand(n,7)==6, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Formula

Sum_{n>=2} 1/a(n) = 5.126608057149204485684180689064467269298250594297584060475240185531109866051... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022