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A209615
Completely multiplicative with a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e otherwise.
11
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, 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, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1
OFFSET
1,1
COMMENTS
Turn sequence of the alternate paperfolding curve. Davis and Knuth define the alternate paperfolding curve by folding a long strip of paper repeatedly in half alternately to the left side or right side, then unfolding it so each crease is 90 degrees (or other angle). a(n) is their d(n) at equation 4.2. Their equation 6.2 (varied to d(2) = -1 as described there) is equivalent to the definition here. The curve is drawn by a unit step forward, turn a(1)*90 degrees left, a unit step forward, turn a(2)*90 degrees left, and so on. - Kevin Ryde, Apr 18 2020
REFERENCES
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, CSLI Publications, 2010, pages 571-614.
FORMULA
G.f.: Sum_{k>=0} (-1)^k * x^(2^k) / (1 + x^(2^(k+1))).
G.f. A(x) satisfies A(x) + A(x^2) = x / (1 + x^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v + w) - (u + v)^2 * (1 + 2*(v + w)).
If p is prime then a(p) = 1 if and only if p is in A002144.
a(4*n + 1) = 1, a(4*n + 3) = -1. a(2*n) = a(3*n) = a(-n) = -a(n).
a(n) = -(-1)^A106665(n-1) unless n=0.
a(2n) = -a(n), a(2n+1) = (-1)^n. [Davis and Knuth equation 4.2] - Kevin Ryde, Apr 18 2020
From Jianing Song, Apr 24 2021: (Start)
a(n) = 1 <=> A003324(n) = 1 or 4, a(n) = -1 <=> A003324(n) = 2 or 3. In other words, a(n) = Legendre(A003324(n), 5) == A003324(n)^2 (mod 5).
a(n) = A034947(n) * (-1)^(v2(n)), where v2(n) = A007814(n) is the 2-adic valuation of n.
Dirichlet g.f.: beta(s)/(1 + 2^(-s)). (End)
EXAMPLE
x - x^2 - x^3 + x^4 + x^5 + x^6 - x^7 - x^8 + x^9 - x^10 - x^11 - x^12 + ...
From Kevin Ryde, Apr 18 2020: (Start)
... alternate
| -1 paperfolding
-1 --->\ \<--- +1 curve
^ -1 | ^
| v | turns +1 left
start --> +1 +1 ---> +1 or -1 right
(End)
PROG
(PARI) {a(n) = my(v); if( n==0, 0, v = valuation( n, 2); (-1)^(n/2^v\2 + v))};
(PARI) {a(n) = if( n!=0, -kronecker( -1, n) * (-1)^if( n!=0, 1 - valuation( n, 2) %2))};
(PARI) {a(n) = my(A, p, e, f); sign(n) * if( n==0, 0, A = factor(abs(n)); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; (-1)^(e * (p%4 != 1))) )};
CROSSREFS
Indices of 1: A338692 = A016813 U A343501; indices of -1: A338691 = A004767 U A343500.
Inverse Moebius transform gives A338690.
Sequence in context: A262725 A070748 A154990 * A242179 A319117 A063747
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Mar 10 2012
STATUS
approved