A106665 Alternate paper-folding (or alternate dragon curve) sequence.

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

Offset: 0

Author

Duane K. Allen (computeruser(AT)sprintmail.com), May 13 2005

Keywords

• easy
• nonn

Comments

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

Examples

			1 + x^3 + x^4 + x^5 + x^8 + x^12 + x^13 + x^15 + x^16 + x^19 + x^20 + ...
		

References

• 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)

Links

• Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.3.2 "The alternate paper-folding sequence", pp. 90-92
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves - II, Journal of Recreational Mathematics, Vol. 3, No. 3, 1970, pp. 133-149, see section 4. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, 2011, pages 571-614.
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
• 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)
• Kevin Ryde, Iterations of the Alternate Paperfolding Curve, see index "TurnLpred" with a(n) = TurnLpred(n+1).
• Wikipedia, Dragon curve

Crossrefs

Cf. A014577, A209615.

Cf. A034947, A096268.

Programs

Maple

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

Mathematica

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 *)

PARI

{a(n) = n++; if( n==0, 0, v = valuation( n, 2); (n/2^v\2 + v+1) %2 )} /* Michael Somos, Mar 10 2012 */

Python

def A106665(n): return ((n+1>>(m:=(~(n+1)&n).bit_length()))+1>>1)+m&1 # Chai Wah Wu, Feb 25 2025

Formula

For n >= 0, a(4n) = 1, a(4n+2) = 0, a(2n+1) = 1 - a(n).

(-1)^a(n) = -A034947(n+1) * (-1)^A096268(n). - Alec Edgington (alec(AT)obtext.com), Aug 02 2010

-(-1)^a(n) = A209615(n+1). - Michael Somos, Mar 10 2012

G.f.: 1/(1 - x^4) + Sum_{k>=1} x^(2*((-1 - (-2)^(k - 1)) mod 2^(k + 1)) + 1)/(1 - x^(2^(k + 2))). - Miles Wilson, Nov 18 2024

Extensions

Edited by N. J. A. Sloane, Jun 04 2010 to include material from discussions on the Sequence Fans Mailing List.