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%I #98 Sep 17 2022 14:29:03
%S 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,
%T 53,54,55,58,59,61,62,63,85,86,87,90,91,93,94,95,106,107,109,110,111,
%U 117,118,119,122,123,125,126,127,170,171,173,174,175,181
%N Numbers with no adjacent 0's in binary expansion.
%C Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
%C 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]
%C 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. See the corresponding example below. - _Gus Wiseman_, Apr 04 2020
%C The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - _Peter Munn_, Sep 06 2022
%H Indranil Ghosh, <a href="/A003754/b003754.txt">Table of n, a(n) for n = 1..50000</a> (terms 1..1000 from T. D. Noe)
%H J.-P. Allouche, J. Shallit and G. Skordev, <a href="https://doi.org/10.1016/j.disc.2004.12.004">Self-generating sets, integers with missing blocks and substitutions</a>, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
%H Robert Baillie and Thomas Schmelzer, <a href="https://library.wolfram.com/infocenter/MathSource/7166/">Summing Kempner's Curious (Slowly-Convergent) Series</a>, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
%H David Garth and Adam Gouge, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Garth/garth14.html">Affinely Self-Generating Sets and Morphisms</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
%H Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H Tomi Kärki, Anne Lacroix, and Michel Rigo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Rigo/rigo6.html">On the recognizability of self-generating sets</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ahnentafel">Ahnentafel</a>.
%H Witzel, Stefan <a href="https://doi.org/10.1007/s10711-017-0247-8">On panel-regular ~A2 lattices</a> Geom. Dedicata 191, 85-135 (2017).
%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%F Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - _Amiram Eldar_, Feb 12 2022
%e 21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
%e From _Gus Wiseman_, Apr 04 2020: (Start)
%e The terms together with the corresponding compositions begin:
%e 0: () 30: (1,1,1,2) 90: (2,1,2,2)
%e 1: (1) 31: (1,1,1,1,1) 91: (2,1,2,1,1)
%e 2: (2) 42: (2,2,2) 93: (2,1,1,2,1)
%e 3: (1,1) 43: (2,2,1,1) 94: (2,1,1,1,2)
%e 5: (2,1) 45: (2,1,2,1) 95: (2,1,1,1,1,1)
%e 6: (1,2) 46: (2,1,1,2) 106: (1,2,2,2)
%e 7: (1,1,1) 47: (2,1,1,1,1) 107: (1,2,2,1,1)
%e 10: (2,2) 53: (1,2,2,1) 109: (1,2,1,2,1)
%e 11: (2,1,1) 54: (1,2,1,2) 110: (1,2,1,1,2)
%e 13: (1,2,1) 55: (1,2,1,1,1) 111: (1,2,1,1,1,1)
%e 14: (1,1,2) 58: (1,1,2,2) 117: (1,1,2,2,1)
%e 15: (1,1,1,1) 59: (1,1,2,1,1) 118: (1,1,2,1,2)
%e 21: (2,2,1) 61: (1,1,1,2,1) 119: (1,1,2,1,1,1)
%e 22: (2,1,2) 62: (1,1,1,1,2) 122: (1,1,1,2,2)
%e 23: (2,1,1,1) 63: (1,1,1,1,1,1) 123: (1,1,1,2,1,1)
%e 26: (1,2,2) 85: (2,2,2,1) 125: (1,1,1,1,2,1)
%e 27: (1,2,1,1) 86: (2,2,1,2) 126: (1,1,1,1,1,2)
%e 29: (1,1,2,1) 87: (2,2,1,1,1) 127: (1,1,1,1,1,1,1)
%e (End)
%p 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:
%p 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:
%p # _R. J. Mathar_, Oct 23 2010
%t Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {___, 0, 0, ___}]&] (* _Jean-François Alcover_, Oct 25 2011 *)
%t 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 *)
%o (Haskell)
%o a003754 n = a003754_list !! (n-1)
%o a003754_list = filter f [0..] where
%o f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
%o -- _Reinhard Zumkeller_, Dec 07 2012, Oct 19 2011
%o (PARI) is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ _Charles R Greathouse IV_, Feb 06 2017
%o (Python)
%o i=0
%o while i<=500:
%o if "00" not in bin(i)[2:]:
%o print(str(i), end=',')
%o i+=1 # _Indranil Ghosh_, Feb 11 2017
%Y 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); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - _Reinhard Zumkeller_, May 25 2005
%Y Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
%Y Complement of A004753.
%Y Cf. A023705, A196168, A280873.
%Y Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
%Y Cf. A318928.
%K nonn,easy,base,nice
%O 1,3
%A _N. J. A. Sloane_
%E Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - _Gus Wiseman_, Apr 04 2020
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