A004742 Numbers whose binary expansion does not contain 101.

0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 19, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 48, 49, 50, 51, 56, 57, 60, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 76, 78, 79, 96, 97, 98, 99, 100, 102, 103, 112, 113, 114, 115, 120, 121, 124, 126, 127

Offset

1,3

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.

Formula

Sum_{n>=2} 1/a(n) = 6.198475910942069028389983717965787117743378665090593775808705963863146498248... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

Mathematica

Select[Range[0, 130], !StringContainsQ[IntegerString[#, 2], "101"] &] (* Amiram Eldar, Feb 13 2022 *)

Prog

(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>4, if(bitand(n, 7)==5, return(0)); n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017
(PARI) is(n)=!bitand(bitand(n, n>>2), bitneg(n>>1)) \\ Charles R Greathouse IV, Oct 28 2021
(PARI) searchLE(S, x)=my(t=setsearch(S, x)); if(t, t, setsearch(S, x, 1)-1); \\ finds last element <= x
expand(~v, lim)=my(b=exponent(v[#v]+1), B=1<<b, mx=searchLE(v, lim-B), block1=setsearch(v, B\4-1), block2=setsearch(v, B\2)); for(i=1, min(block1, mx), listput(v, B+v[i])); for(i=block2, mx, listput(v, B+v[i])); if(v[#v]>lim, listpop(~v));
list(lim)=lim\=1; if(lim<5, return(if(lim<0, [], [0..lim]))); my(v=List([0..3])); for(b=3, exponent(lim+1), expand(~v, 2^b-1)); expand(~v, lim); Vec(v)
(Haskell)
a004742 n = a004742_list !! (n-1)
a004742_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 5 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Crossrefs

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004743 (no 110), A003726 (no 111).

Sequence in context: A366976 A031998 A023759 * A277817 A336231 A100290

Adjacent sequences: A004739 A004740 A004741 * A004743 A004744 A004745

Keyword

nonn,base,easy

Author

N. J. A. Sloane

Status

approved