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A003754 Numbers with no 2 adjacent 0's in binary expansion. 25
0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1)-a(n); A104326(n) = A007088(a(n)); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005

LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)

J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.

David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Article 07.1.5, 10 (2007) 1-13.

T. Karki, A. Lacroix, M. Rigo, On the recognizability of self-generating sets, JIS 13 (2010) #10.2.2.

Index entries for 2-automatic sequences.

EXAMPLE

21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - Indranil Ghosh, Feb 11 2017

MAPLE

isA003754 := proc(n) local bdgs ; bdgs := convert(n, base, 2) ; for i from 2 to nops(bdgs) do if op(i, bdgs)=0 and op(i-1, bdgs)= 0 then return false; end if; end do; return true; end proc:

A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:

# R. J. Mathar, Oct 23 2010

MATHEMATICA

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {___, 0, 0, ___}]&] (* Jean-Fran├žois Alcover, Oct 25 2011 *)

Select[Range[0, 200], SequenceCount[IntegerDigits[#, 2], {0, 0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

PROG

(Haskell)

a003754 n = a003754_list !! (n-1)

a003754_list = filter f [0..] where

   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)

-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

(PARI) is(n)=n=bitor(n, n>>1)+1; n>>=valuation(n, 2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

(Python)

i=0

j=1

while j<=50000:

....if bin(i)[2:].count("00")==0:

........print str(j)+" "+str(i)

........j+=1

....i+=1 # Indranil Ghosh, Feb 11 2017

CROSSREFS

Complement of A004753.

Cf. A023705, A196168.

Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).

Sequence in context: A087007 A047586 A103841 * A293427 A087006 A235991

Adjacent sequences:  A003751 A003752 A003753 * A003755 A003756 A003757

KEYWORD

nonn,easy,base,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 17 10:50 EDT 2017. Contains 293469 sequences.