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 A106665 Alternate paper-folding (or alternate dragon curve) sequence. 4
 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; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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 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. 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 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 EXAMPLE 1 + x^3 + x^4 + x^5 + x^8 + x^12 + x^13 + x^15 + x^16 + x^19 + x^20 + ... 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 *) PROG (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 */ CROSSREFS Cf. A014577, A209615. Cf. A034947, A096268. Sequence in context: A051341 A057211 A120531 * A004609 A287674 A071001 Adjacent sequences:  A106662 A106663 A106664 * A106666 A106667 A106668 KEYWORD easy,nonn AUTHOR Duane K. Allen (computeruser(AT)sprintmail.com), May 13 2005 EXTENSIONS Edited by N. J. A. Sloane, Jun 04 2010 to include material from discussions on the Sequence Fans Mailing List. STATUS approved

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Last modified October 2 19:28 EDT 2022. Contains 357228 sequences. (Running on oeis4.)