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A014577
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The regular paper-folding (or dragon curve) sequence.
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24
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1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0
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OFFSET
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0,1
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COMMENTS
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Can be computed by storing only one large integer. It is the complement of the bit to the left of the least significant "1" in the binary expansion of n. E.g. n = 4 = 100, so a(4) = complement of bit to left of 1, = 1. - Bob Brown (bobb(AT)webaccess.net), Nov 28 2001
To construct the sequence : start from 1,(..),0,(..),1,(..),0,(..),1,(..),0,(..),1,(..),0,... and fill undefined places with the sequence itself. - Benoit Cloitre, Jul 08 2007
A014577 is a generator for A088748: begin A088748 with "1" then add 1 if A014577: (1, 1, 0, 1, 1,...) = 1; subtract 1 otherwise, getting (1, 2, 3, 2,...). [Gary W. Adamson, Aug 30 2009]
Contribution from Gary W. Adamson, Apr 11 2010: (Start)
After changing the offset to 1: (1, 1, 0, 1, 1, 0, 0, 1, 1, 1,...) = the
characteristic function of A091072: (1, 2, 4, 5, 8, 9, 10, 13,...). (End)
Turns (by 90 degrees) of the Heighway dragon which can be rendered as follows: [Init] Set n=0 and direction=0. [Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively. [Next] Set n=n+1 and goto (draw). See fxtbook link below. [Joerg Arndt, Apr 15 2010]
Sequence can be obtained by L-system with rules L->L1R, R->L0R, 1->1, 0->0, starting with L, and dropping all L and R (see example). [Joerg Arndt, Aug 28 2011]
Comment from Gary W. Adamson, Jun 20 2012. (Start)
One half of the infinite Farey Tree can be mapped one-to-one onto A014577 since both sequences can be derived directly from the binary. First few terms are:
1,...1,...0,...1,...1,...0,...0,...1,...1,...1,...
1/2.2/3..1/3..3/4..3/5..2/5..1/4..4/5..5/7..5/8,..
Infinite Farey Tree fractions can be derived from the binary by appending a repeat of rightmost binary term to the right, then recording the number of runs to obtain the continued fraction representation. Example: 9 = 1001 which becomes 10011 which becomes [1,2,2] = 5/7. (End)
Contribution from Gary W. Adamson, Jun 24 2012: (Start) The sequence can be considered as a binomial transform operator for a target sequence S(n). Replace the first 1 in A014577 with the first term in S(n), then for successive "1" term in A014577, map the next higher term in S(n). If "0" in A014577, map the next lower term in S(n). Using the sequence S(n) = (1, 3, 5, 7,...), we obtain (1), (3, 1), (3, 5, 3, 1), (3, 5, 7, 5, 3, 5, 3, 1),.... Then parse the terms into subsequences of 2^k terms, adding the terms in each string. We obtain (1, 4, 12, 32, 80,...), the binomial transform of (1, 3, 5, 7,...). The 8 bit string has one 1, three 5's, three 7's and one 1) as expected, or (1, 3, 3, 1) dot (1, 3, 5, 7). (End)
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 155, 182.
M. Gardner, Mathematical Magic Show. New York: Vintage, pp. 207-209 and 215-220, 1978.
G. Melancon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
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LINKS
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Table of n, a(n) for n=0..98.
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.
Joerg Arndt, Fxtbook, p.88-92, image of the dragon curve on p. 89
A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 63. Book's website
J. E. S. Socolar and J. M. Taylor, An aperiodic hexagonal tile
Eric Weisstein's World of Mathematics, Dragon curve.
Index entries for sequences obtained by enumerating foldings
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FORMULA
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a(n) = (1+jacobi(-1,n))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012
Set a=1, b=0, S(0)=a, S(n+1) = S(n),a,F(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).
a(4*n) = 1, a(4*n+2) = 0, a(2*n+1) = a(n). a(n) = 1 - A014707(n) = 2 - A014709(n) = A014710(n) - 1. [Ralf Stephan, Jul 03 2003]
Set a=1, b=0 S(0)=a, S(n+1) =S(n),a,M(S(n)), where M(S) is S but the bit in the middle position flicked. (Proof via isomorphism of both formulae to a modified string substitution.) [Benjamin Heiland, Dec 11 2011]
Can be generated directly from A005811:
1,...2,...1,...2,...3,...2,...1,...2,...3,... = A005811.
1,...1,...0,...1,...1,...0,...0,...1,...1,... = A014577.
By inspection, A014577 = 1 if the corresponding term in A005811 is greater than the previous A005811 term; else 0. - Gary W. Adamson, Jun 20 2012.
a((2*n+1)*2^p-1) = (n+1) mod 2, p >= 0. - Johannes W. Meijer, Jan 28 2013
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EXAMPLE
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1 + x + x^3 + x^4 + x^7 + x^8 + x^9 + x^12 + x^15 + x^16 + x^17 + x^19 + ...
From Joerg Arndt, Aug 28 2011: (Start)
Generation via string substitution:
Start: L
Rules:
L --> L1R
R --> L0R
0 --> 0
1 --> 1
-------------
0: (#=1)
L
1: (#=3)
L1R
2: (#=7)
L1R1L0R
3: (#=15)
L1R1L0R1L1R0L0R
4: (#=31)
L1R1L0R1L1R0L0R1L1R1L0R0L1R0L0R
5: (#=63)
L1R1L0R1L1R0L0R1L1R1L0R0L1R0L0R1L1R1L0R1L1R0L0R0L1R1L0R0L1R0L0R
Drop all L and R to obtain 1101100111001001110110001100100
(End)
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MAPLE
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nmax:=98: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := (n+1) mod 2 od: od: seq(a(n), n=0..nmax); # [Johannes W. Meijer, Jan 28 2013]
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MATHEMATICA
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Boole[ EvenQ[ (#/2^IntegerExponent[#, 2] - 1)/2]] & /@ Range[99] (* Jean-François Alcover, Feb 16 2012, after Gary W. Adamson *)
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PROG
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(C++) /* code from the fxt library, about 5 CPU cycles per computation */
bool bit_paper_fold(ulong k)
{
ulong h = k & -k; /* == lowest_one(k) */
k &= (h<<1);
return ( k==0 );
} /* Joerg Arndt, Apr 15 2010 */
(PARI) {a(n) = if( n%2, a(n\2), 1 - (n/2%2))} /* Michael Somos, Feb 05 2012 */
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CROSSREFS
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The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012
Cf. A038189, A059125, A065339, A005811, A220466
A082410(n+2)=a(n).
Cf. A088748 [Gary W. Adamson, Aug 30 2009]
Cf. A091072 [Gary W. Adamson, Apr 11 2010]
Sequence in context: A100672 A079559 A175480 * A157926 A131377 A077049
Adjacent sequences: A014574 A014575 A014576 * A014578 A014579 A014580
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Eric W. Weisstein
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EXTENSIONS
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More terms from Ralf Stephan, Jul 03 2003
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STATUS
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approved
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