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%I #40 Jan 03 2023 09:21:57
%S 1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,-1,1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,
%T -1,1,-1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,-1,1,1,1,-1,1,1,1,-1,1,-1,1,-1,
%U 1,1,1,-1,1,1,1,-1,1,1,1,-1,1,-1,1,-1,1,1,1,-1,1,1,1,-1,1,1,1,-1,1,-1,1,-1
%N Period-doubling turn sequence, +1 when the 2-adic valuation of n is even or -1 when odd.
%C a(n)=+1 when the number of low 0 bits of n is even, and a(n)=-1 when odd. This is the "period doubling" sequence A096268 but using +1,-1. See A096268 and its complement A035263 for more. The number of low 0 bits of n is A007814. a(n) is completely multiplicative since A007814(n*m) = A007814(n) + A007814(m).
%C a(n) is among some completely multiplicative +1,-1 sequences considered by Davis and Knuth for forming curves by unfolding. a(n) is their d(n) at equation 6.4. The curve can be drawn by successively going forward a unit step and turning by a(n)*angle. Their "bending" angle T is equivalent to turns by 180-T degrees. They draw bending angle 90 degrees which is merely 4 unit squares repeatedly traversed; and bending 60 degrees "Fido" and 120 degrees which are bigger and more interesting. Partial sums A068639 are the directions (net total turn) of the segments.
%D Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, 2010, pages 571-614.
%H Kevin Ryde, <a href="/A309873/b309873.txt">Table of n, a(n) for n = 1..4096</a>
%H Kevin Ryde, <a href="/A309873/a309873.pdf">Sample Images</a> (and <a href="/A309873/a309873.tex">LaTeX source</a>)
%H Kevin Ryde, <a href="/A309873/a309873.l.txt">Xfractint L-System Specification</a>
%F a(n) = (-1)^A007814(n) = -(-1)^A065882(n) = 1 - 2*A096268(n-1) = 2*A035263(n) - 1 = A035263(n) - A096268(n-1).
%F a(n) = A068639(n) - A068639(n-1).
%F a(A003159) = 1, a(A036554) = -1.
%F For prime p, a(p) = -1 if p=2 (even), a(p) = 1 if p odd [Davis and Knuth, which together with completely multiplicative defines a(n)].
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/3. - _Amiram Eldar_, Sep 18 2022
%F Dirichlet g.f.: zeta(s)*(2^s-1)/(2^s+1). - _Amiram Eldar_, Jan 03 2023
%t Array[(-1)^IntegerExponent[#, 2] &, 100] (* _Amiram Eldar_, Aug 22 2019 *)
%o (PARI) a(n) = (-1)^valuation(n,2);
%o (UCB Logo) ; a(n), and draw "Fido" of Davis and Knuth
%o to a :n
%o output 2*(remainder (bitxor :n :n-1) 3) - 1
%o end
%o setheading 90 ; start East
%o repeat 4095 [ forward 7 ; pixels
%o left (a repcount) * 120 ] ; or try 60 or 90
%o (Python)
%o def A309873(n): return -1 if (~n & n-1).bit_length()&1 else 1 # _Chai Wah Wu_, Dec 26 2022
%Y Cf. A096268, A035263, A003159, A036554, A007814.
%Y Partial sums A068639.
%K mult,sign,easy
%O 1
%A _Kevin Ryde_, Aug 21 2019
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