Search: id:A010060 Results 1-1 of 1 results found. %I A010060 %S A010060 0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0, %T A010060 1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1, %U A010060 1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,1 %N A010060 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's. %C A010060 The sequence is cube-free (does not contain three consecutive identical blocks) and is overlap-free (does not contain XYXYX where X is 0 or 1 and Y is any string of 0's and 1's). %C A010060 a(n) = "parity sequence" = parity of number of 1's in binary representation of n. %C A010060 To construct the sequence: alternate blocks of 0's and 1's of successive lengths A003159(k)-A003159(k-1), k=1,2,3,... (A003159(0)=0). Example: since the first seven differences of A003159 are 1,2,1,1,2,2,2, the sequence starts with 0,1,1,0,1,0,0,1,1,0,0. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 10 2003 %C A010060 Characteristic function of A000069 (odious numbers). a(n) = 1-A010059(n) = A001285(n)-1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), Jun 20 2003 %C A010060 a(n)=S2(n)mod 2, where S2(n) = sum of digits of n, n in base-2 notation. There is a class of generalized Thue-Morse sequences : Let Sk(n) = sum of digits of n; n in base-k notation. Let F(t) be some arithmetic function. Then a(n)= F(Sk(n)) mod m is a generalised Thue-Morse sequence. The classical Thue-Morse sequence is the case k=2, m=2, F(t)= 1. - Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), Feb 12 2008 %C A010060 More generally, the partial sums of the generalized Thue-Morse sequences a(n)=F(Sk(n)) mod m are fractal, where Sk(n) is sum of digits of n, n in base k; F(t) is an arithmetic function; m integer. - Ctibor O. ZIZKA (ctibor.zizka(AT)seznam.cz), Feb 25 2008 %C A010060 Starting with offset 1, = running sums mod 2 of the kneading sequence (A035263, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1,...); also parity of A005187: (1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2008 %C A010060 Generalized Thue-Morse sequences mod n (n>1) = the array shown in A141803. As n -> Inf. the sequences -> (1, 2, 3,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 10 2008 %C A010060 The Thue-Morse sequence for N = 3 = A053838, (sum of digits of n in base 3, mod 3): (0, 1, 2, 1, 2, 0, 2, 0, 1, 1, 2,...) = A004128 mod 3. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 24 2008] %C A010060 For all positive integers k, the subsequence a(0) to a(2^k-1) is identical to the subsequence a(2^k+2^(k-1)) to a(2^(k+1)+2^(k-1)-1). That is to say, the first half of A_k is identical to the second half of B_k, and the second half of A_k is identical to the first quarter of B_{k+1}, which consists of the k/2 terms immediately following B_k. %C A010060 Proof: The subsequence a(2^k+2^(k-1)) to a(2^(k+1)-1), the second half of B_k, is by definition formed from the subsequence a(2^(k-1)) to a(2^k-1), the second half of A_k, by interchanging its 0s and 1s. In turn, the subsequence a(2^(k-1)) to a(2^k-1), the second half of A_k, which is by definition also B_{k-1}, is by definition formed from the subsequence a(0) to a(2^(k-1)-1), the first half of A_k, which is by definition also A_{k-1}, by interchanging its 0s and 1s. Interchanging the 0s and 1s of a subsequence twice leaves it unchanged, so the subsequence a(2^k+2^(k-1)) to a(2^(k+1)-1), the second half of B_k, must be identical to the subsequence a(0) to a(2^(k-1)-1), the first half of A_k. %C A010060 Also, the subsequence a(2^(k+1)) to a(2^(k+1)+2^(k-1)-1), the first quarter of B_{k+1}, is by definition formed from the subsequence a(0) to a(2^(k-1)-1), the first quarter of A_{k+1}, by interchanging its 0s and 1s. As noted above, the subsequence a(2^(k-1)) to a(2^k-1), the second half of A_k, which is by definition also B_{k-1}, is by definition formed from the subsequence a(0) to a(2^(k-1)-1), which is by definition A_{k-1}, by interchanging its 0s and 1s, as well. If two subsequences are formed from the same subsequence by interchanging its 0s and 1s then they must be identical, so the subsequence a(2^(k+1)) to a(2^(k+1)+2^(k-1)-1), the first quarter of B_{k+1}, must be identical to the subsequence a(2^(k-1)) to a(2^k-1), the second half of A_k. %C A010060 Therefore the subsequence a(0),..., a(2^(k-1)-1), a(2^(k-1)),..., a(2^k-1) is identical to the subsequence a(2^k+2^(k-1)),..., a(2^(k+1)-1), a(2^(k+1)),..., a(2^(k+1)+2^(k-1)-1), QED. %D A010060 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29. %D A010060 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15. %D A010060 F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10). %D A010060 J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228. %D A010060 F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99. %D A010060 S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154. %D A010060 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.8. %D A010060 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105. %D A010060 J. Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429. %D A010060 G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150. %D A010060 A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159. %D A010060 B. Kitchens, Review of "Computational Ergodic Theory" by G. H. Choe, Bull. Amer. Math. Soc., 44 (2007), 147-155. %D A010060 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23. %D A010060 M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100. %D A010060 C. A. Pickover, Wonders of Numbers, Adventures in Mathematics, Mind and Meaning, Chapter 17, 'The Pipes of Papua,' Oxford University Press, Oxford, England, 2000, pages 34 - 38. %D A010060 C. A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60. %D A010060 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 316. %D A010060 Benoit Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, Quand les maths se font discretes, Le Pommier, 2008 (ISBN 978-2-7465-0370-0). %D A010060 A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6. %D A010060 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 890. %H A010060 N. J. A. Sloane, Table of n, a(n) for n = 0..16383 %H A010060 Joerg Arndt, Fxtbook %H A010060 J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461. %H A010060 J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences. %H A010060 J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II %H A010060 J.-P. Allouche and J. O. Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16. %H A010060 Ricardo Astudillo, On a Class of Thue-Morse Type Sequences, J. Integer Seqs., Vol. 6, 2003. %H A010060 Jean Berstel, Home Page %H A010060 E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215. %H A010060 A. S. Fraenkel, Home Page %H A010060 A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. %H A010060 Michael Gilleland, Some Self-Similar Integer Sequences %H A010060 M. Morse, Recurrent geodesics on a surface of negative curvature (page images), Trans. Amer. Math. Soc., 22 (1921), 84-100. %H A010060 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review %H A010060 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (1). %H A010060 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2). %H A010060 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3). %H A010060 Index entries for "core" sequences %H A010060 Index entries for characteristic functions %F A010060 a(2n)=a(n), a(2n+1)=1-a(n), a(0)=0. Also, a(k+2^m)=1-a(k) if 0<=k<2^m. %F A010060 Let S(0) = 0 and for k >=1, construct S(k) from S(k-1) by mapping 0 -> 01 and 1 -> 10; sequence is S(infinity). %F A010060 G.f.: (1/(1-x) - product_{k>=0} 1-x^(2^k))/2. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 23 2003 %F A010060 a(0)=0, a(n)=(n+a(floor(n/2))) mod 2; also a(0)=0, a(n)=(n-a(floor(n/ 2))) mod 2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 10 2003 %F A010060 a(n)=-1+sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=-1+A001316(n) {mod 3} - Benoit Cloitre (benoit7848c(AT)orange.fr), May 09 2004 %F A010060 Let b(1)=1 and b(n)=b(ceil(n/2))-b(floor(n/2)) then a(n-1)=(1/2)*(1-b(2n-1)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 26 2005 %F A010060 a(n) = A001969(n) - 2n. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006 %F A010060 a(n) = A115384(n) - A115384(n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 26 2007 %F A010060 For n>=0, a(A004760(n+1))=1-a(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 25 2009] %F A010060 a(A160217(n))=1-a(n). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 05 2009] %e A010060 The evolution starting at 0 is: %e A010060 .0 %e A010060 .0, 1 %e A010060 .0, 1, 1, 0 %e A010060 .0, 1, 1, 0, 1, 0, 0, 1 %e A010060 .0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 %e A010060 .0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1 %e A010060 ....... %e A010060 A_2 = 0 1 1 0, so B_2 = 1 0 0 1 and A_3 = A_2 B_2 = 0 1 1 0 1 0 0 1. %p A010060 s := proc(k) local i, ans; ans := [ 0,1 ]; for i from 0 to k do ans := [ op(ans),op(map(n->(n+1) mod 2, ans)) ] od; RETURN(ans); end; t1 := s(6); A010060 := n->t1[n]; # s(k) gives first 2^(k+2) terms. %p A010060 a := proc(k) b := [0]: for n from 1 to k do b := subs({0=[0,1], 1=[1, 0]},b) od: b; end; # a(k), after the removal of the brackets, gives the first 2^k terms. # Example: a(3); gives [[[[0, 1], [1, 0]], [[1, 0], [0, 1]]]] %p A010060 a:=proc(n) local n2: n2:=convert(n,base,2): sum(n2[j],j=1..nops(n2)) mod 2; end: seq(a(n),n=0..104); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 19 2005 %t A010060 Table[ If[ OddQ[ Count[ IntegerDigits[n, 2], 1]], 1, 0], {n, 0, 100}]; %t A010060 mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt] ], {n, 0, 6} ]; Prepend[ RealDigits[ N[ ToExpression[mt], 2^7] ] [ [1] ], 0] %t A010060 Mod[ Count[ #, 1 ]& /@Table[ IntegerDigits[ i, 2 ], {i, 0, 2^7 - 1} ], 2 ] (from Harlan J. Brothers, Feb 05 2005) %t A010060 Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 7] (* Robert G. Wilson v Sep 26 2006) %o A010060 (Haskell) a = 0: interleave (complement a) (tail a) where {complement = map (1 - ); interleave (x:xs) ys = x: interleave ys xs} (from Doug McIlroy (doug(AT)cs.dartmouth.edu), Jun 29 2003) %o A010060 (PARI) a(n)=if(n<1,0,sum(k=0,length(binary(n))-1,bittest(n,k))%2) %o A010060 (PARI) a(n)=if(n<1,0,subst(Pol(binary(n)), x,1)%2) %o A010060 (PARI) { default(realprecision, 6100); x=0.0; m=20080; for (n=1, m-1, x=x+x; x=x+sum(k=0, length(binary(n))-1, bittest(n, k))%2); x=2*x/ 2^m; for (n=0, 20000, d=floor(x); x=(x-d)*2; write("b010060.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 28 2009] %Y A010060 Cf. A001285 (for 1, 2 version), A010059 (1, 0 version), A048707. A010060(n)=A000120(n) mod 2. %Y A010060 Cf. A080813, A080814, A036581, A108694. See also the Thue (or Roth) constant A014578. %Y A010060 Cf. also A001969, A035263, A005187, A115384, A132680, A141803, A104248. %Y A010060 Backward first differences give A029883. %Y A010060 A004128, A053838 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 24 2008] %Y A010060 Cf. A059448. %Y A010060 Sequence in context: A053866 A156595 A143222 this_sequence A118247 A122257 A129950 %Y A010060 Adjacent sequences: A010057 A010058 A010059 this_sequence A010061 A010062 A010063 %K A010060 nonn,core,easy,nice %O A010060 0,1 %A A010060 N. J. A. Sloane (njas(AT)research.att.com). %E A010060 Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 29 2000 Search completed in 0.004 seconds