%I #61 Dec 15 2021 14:36:16
%S 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,
%T 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,
%U 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
%N Alternate paper-folding (or alternate dragon curve) sequence.
%C 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.
%C 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
%D M. Gardner, "The Dragon Curve and Other Problems (Mathematical Games)", Scientific American, 1967, columns for March, April, July.
%D M. Gardner, "Mathematical Magic Show" (contains the dragon curve columns).
%D D. E. Knuth, "Art of Computer Programming," vol. 2, 3rd. ed., "Seminumerical > Algorithms," (section 4.1)
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 1.31.3.2 "The alternate paper-folding sequence", pp. 90-92
%H 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, <a href="http://www-cs-faculty.stanford.edu/~uno/fg.html">Selected Papers on Fun and Games</a>, 2011, pages 571-614.
%H William J. Gilbert, <a href="http://www.math.uwaterloo.ca/~wgilbert/Research/MathIntel.pdf">Fractal geometry derived from complex bases</a>, The Mathematical Intelligencer, Volume 4, Number 2 (June 1982), pp. 78-86. (ISSN 0343-6993; DOI 10.1007/BF03023486)
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/alternate/index.html">Iterations of the Alternate Paperfolding Curve</a>, see index "TurnLpred" with a(n) = TurnLpred(n+1).
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dragon_curve">Dragon curve</a>
%F For n >= 0, a(4n) = 1, a(4n+2) = 0, a(2n+1) = 1 - a(n).
%F (-1)^a(n) = -A034947(n+1) * (-1)^A096268(n). - Alec Edgington (alec(AT)obtext.com), Aug 02 2010
%F -(-1)^a(n) = A209615(n+1). - _Michael Somos_, Mar 10 2012
%e 1 + x^3 + x^4 + x^5 + x^8 + x^12 + x^13 + x^15 + x^16 + x^19 + x^20 + ...
%p a:= proc(n) option remember;
%p `if`(irem(n, 4)=0, 1, `if`(irem(n, 2)=1, 1-a((n-1)/2), 0))
%p end:
%p seq(a(n), n=0..120); # _Alois P. Heinz_, Mar 10 2012
%t a[n_] := a[n] = Switch[Mod[n, 4], 0, 1, 2, 0, _, 1-a[(n-1)/2]];
%t Table[a[n], {n, 0, 120}] (* _Jean-François Alcover_, Mar 15 2017 *)
%o (C++) /* "ulong" stands for "unsigned long" */
%o bool bit_paper_fold_alt(ulong k)
%o {
%o ulong h = k & -k; // == lowest_one(k)
%o h <<= 1;
%o ulong t = h & (k ^ 0xaaaaaaaaUL); // 32-bit version
%o return ( t!=0 );
%o } /* _Joerg Arndt_, Jun 03 2010 */
%o (PARI) {a(n) = n++; if( n==0, 0, v = valuation( n, 2); (n/2^v\2 + v+1) %2 )} /* _Michael Somos_, Mar 10 2012 */
%Y Cf. A014577, A209615.
%Y Cf. A034947, A096268.
%K easy,nonn
%O 0,1
%A Duane K. Allen (computeruser(AT)sprintmail.com), May 13 2005
%E Edited by _N. J. A. Sloane_, Jun 04 2010 to include material from discussions on the Sequence Fans Mailing List.
|