A003754 - OEIS

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A003754

Numbers with no adjacent 0's in binary expansion.

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.` `Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]` `From Gus Wiseman, Apr 04 2020: (Start)` `The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. For example, the terms together with the corresponding compositions begin:` `0: () 30: (1,1,1,2) 90: (2,1,2,2)` `1: (1) 31: (1,1,1,1,1) 91: (2,1,2,1,1)` `2: (2) 42: (2,2,2) 93: (2,1,1,2,1)` `3: (1,1) 43: (2,2,1,1) 94: (2,1,1,1,2)` `5: (2,1) 45: (2,1,2,1) 95: (2,1,1,1,1,1)` `6: (1,2) 46: (2,1,1,2) 106: (1,2,2,2)` `7: (1,1,1) 47: (2,1,1,1,1) 107: (1,2,2,1,1)` `10: (2,2) 53: (1,2,2,1) 109: (1,2,1,2,1)` `11: (2,1,1) 54: (1,2,1,2) 110: (1,2,1,1,2)` `13: (1,2,1) 55: (1,2,1,1,1) 111: (1,2,1,1,1,1)` `14: (1,1,2) 58: (1,1,2,2) 117: (1,1,2,2,1)` `15: (1,1,1,1) 59: (1,1,2,1,1) 118: (1,1,2,1,2)` `21: (2,2,1) 61: (1,1,1,2,1) 119: (1,1,2,1,1,1)` `22: (2,1,2) 62: (1,1,1,1,2) 122: (1,1,1,2,2)` `23: (2,1,1,1) 63: (1,1,1,1,1,1) 123: (1,1,1,2,1,1)` `26: (1,2,2) 85: (2,2,2,1) 125: (1,1,1,1,2,1)` `27: (1,2,1,1) 86: (2,2,1,2) 126: (1,1,1,1,1,2)` `29: (1,1,2,1) 87: (2,2,1,1,1) 127: (1,1,1,1,1,1,1)` `(End)`
	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., Vol. 292, No. 1-3 (2005), pp. 1-15.` `Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.` `David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.` `Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.` `Wikipedia, Ahnentafel.` `Index entries for 2-automatic sequences.` `Index entries for sequences related to binary expansion of n.`
	FORMULA	`Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022`
	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-François 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` `while i<=500:` `if "00" not in bin(i)[2:]:` `print(str(i), end=', ')` `i+=1 # Indranil Ghosh, Feb 11 2017`
	CROSSREFS	`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` `Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).` `Complement of A004753.` `Cf. A023705, A196168, A280873.` `All of the following pertain to compositions in standard order (A066099):` `- The length is A000120.` `- Compositions without ones are ranked by A022340.` `- The sum is A070939.` `- Compositions with no twos are ranked by A175054.` `- The reverse is A228351.` `- Constant compositions are ranked by A272919.` `- Normal compositions are ranked by A333217.` `- Anti-runs are counted by A333381.` `- Anti-runs are ranked by A333489.` `- Runs-resistance is A333628.` `Cf. A124767, A318928, A329745, A329767.` `Sequence in context: A087007 A047586 A103841 * A293427 A293430 A087006` `Adjacent sequences: A003751 A003752 A003753 * A003755 A003756 A003757`
	KEYWORD	`nonn,easy,base,nice`
	AUTHOR	`N. J. A. Sloane`
	EXTENSIONS	`Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020`
	STATUS	`approved`

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Last modified June 26 19:50 EDT 2022. Contains 354885 sequences. (Running on oeis4.)