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A106665
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Alternate paper-folding (or alternate dragon curve) sequence.
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4
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1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1
(list;
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OFFSET
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0,1
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COMMENTS
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Regular dragon curve (A014577) sequence results from repeated folding of long strip of paper in half in the same direction, say right to left. This alternate dragon curve sequence results from repeated folding of long strip of paper in half in alternating directions, right to left, then left to right and so forth.
In the Wikipedia article "Dragon Curve" note the illustrated description under the heading "[Un]Folding the Dragon" and note that the 1's and 0's in the A106665 description correspond to the L and R folds in the Wikipedia discussion. - Robert Munafo, Jun 03 2010
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REFERENCES
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C. Davis and D. E. Knuth, Number Representations and Dragon Curves - II, Journal of Recreational Mathematics, Vol. 3, No. 3, 1970, pp. 133-149
M. Gardner, "The Dragon Curve and Other Problems (Mathematical Games)", Scientific American, 1967, columns for March, April, July.
M. Gardner, "Mathematical Magic Show" (contains the dragon curve columns).
D. E. Knuth, "Art of Computer Programming," vol. 2, 3rd. ed., "Seminumerical > Algorithms," (section 4.1)
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LINKS
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Table of n, a(n) for n=0..104.
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.3.2 "The alternate paper-folding sequence", pp. 90-92
William J. Gilbert, Fractal geometry derived from complex bases, The Mathematical Intelligencer, Volume 4, Number 2 (June 1982), pp. 78-86 (ISSN 0343-6993; DOI 10.1007/BF03023486)
Wikipedia, Dragon curve
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FORMULA
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For n >= 0, a(4n) = 1, a(4n+2) = 0, a(2n+1) = 1 - a(n).
-(-1)^a(n) = A209615(n+1). - Michael Somos, Mar 10 2012
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EXAMPLE
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1 + x^3 + x^4 + x^5 + x^8 + x^12 + x^13 + x^15 + x^16 + x^19 + x^20 + ...
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MAPLE
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a:= proc(n) option remember;
`if`(irem(n, 4)=0, 1, `if`(irem(n, 2)=1, 1-a((n-1)/2), 0))
end:
seq(a(n), n=0..120); # Alois P. Heinz, Mar 10 2012
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MATHEMATICA
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a[n_] := a[n] = Switch[Mod[n, 4], 0, 1, 2, 0, _, 1-a[(n-1)/2]];
Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Mar 15 2017 *)
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PROG
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(C++) /* "ulong" stands for "unsigned long" */
bool bit_paper_fold_alt(ulong k)
{
ulong h = k & -k; // == lowest_one(k)
h <<= 1;
ulong t = h & (k ^ 0xaaaaaaaaUL); // 32-bit version
return ( t!=0 );
} /* Joerg Arndt, Jun 03 2010 */
(PARI) {a(n) = n++; if( n==0, 0, v = valuation( n, 2); (n/2^v\2 + v+1) %2 )} /* Michael Somos, Mar 10 2012 */
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CROSSREFS
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Cf. A014577, A209615.
(-1)^a(n) = -A034947(n+1) * (-1)^A096268(n). - Alec Edgington (alec(AT)obtext.com), Aug 02 2010
Sequence in context: A051341 A057211 A120531 * A004609 A287674 A071001
Adjacent sequences: A106662 A106663 A106664 * A106666 A106667 A106668
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KEYWORD
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easy,nonn
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AUTHOR
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Duane K. Allen (computeruser(AT)sprintmail.com), May 13 2005
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EXTENSIONS
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Edited by N. J. A. Sloane, Jun 04 2010 to include material from discussions on the Sequence Fans Mailing List.
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STATUS
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approved
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