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A034947 Jacobi (or Kronecker) symbol (-1/n). 22
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, -1, -1, 1, -1, -1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also the regular paper-folding sequence.

For a proof that a(n) equals the paper-folding sequence, see Allouche and Sondow, arXiv v4. - Jean-Paul Allouche and Jonathan Sondow, May 19 2015

It appears that, replacing +1 with 0 and -1 with 1, we obtain A038189. Alternatively, replacing -1 with 0 we obtain (allowing for offset) A014577. - Jeremy Gardiner, Nov 08 2004

Partial sums = A005811 starting (1, 2, 1, 2, 3, 2, 1, 2, 3,...). - Gary W. Adamson, Jul 23 2008

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 155, 182.

H. Cohen, Course in Computational Number Theory, p. 28.

Danielle Cox and K. McLellan, A problem on generation sets containing Fibonacci numbers, Fib. Quart., 55 (No. 2, 2017), 105-113.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

J.-P. Allouche, G.-N. Han, J. Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 2020.

J.-P. Allouche and J. Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, Electron. J. Combin., 22 #1 (2015) P1.59; see p. 8.

J.-P. Allouche and J. Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, arXiv:1408.5770 [math.NT] v4, 2015;  see p. 9.

Jean-Paul Allouche and Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.

Joerg Arndt, Matters Computational (The Fxtbook)

A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.

Eric Weisstein's World of Mathematics, Kronecker Symbol

Index entries for sequences obtained by enumerating foldings

FORMULA

Multiplicative with a(2^e) = 1, a(p^e) = (-1)^(e(p-1)/2) if p>2.

a(2n) = a(n), a(4n+1) = 1, a(4n+3) = -1, a(-n) = -a(n). a(n) = 2*A014577(n-1)-1.

a(prime(n)) = A070750(n) for n > 1 - T. D. Noe, Nov 08 2004

This sequence can be constructed by starting with w = "empty string", and repeatedly applying the map w -> w 1 reverse(-w) [See Allouche and Shallit p. 182). - N. J. A. Sloane, Jul 27 2012

a(n) = (-1)^k, where k is number of primes of the form 4*m + 3 dividing n (counted with multiplicity). - Arkadiusz Wesolowski, Nov 05 2013

Sum(n >= 1, a(n)/n) = Pi/2, due to F. von Haeseler; more generally, sum(n >= 1, a(n)/n^(2d+1)) = Pi^(2d+1)*A000364(d)/(2^(2d+2)-2)(2d)! for d >= 0; see Allouche and Sondow, 2015. - Jean-Paul Allouche and Jonathan Sondow, Mar 20 2015

Dirichlet g.f.: beta(s)/(1-2^(-s)) = L(chi_2(4),s)/(1-2^(-s)). - Ralf Stephan, Mar 27 2015

EXAMPLE

G.f. = x + x^2 - x^3 + x^4 + x^5 - x^6 - x^7 + x^8 + x^9 + x^10 - x^11 - x^12 + ...

MAPLE

with(numtheory): A034947 := n->jacobi(-1, n);

MATHEMATICA

Table[KroneckerSymbol[ -1, n], {n, 0, 100}] (* Corrected by Jean-François Alcover, Dec 04 2013 *)

PROG

(PARI) {a(n) = kronecker(-1, n)};

(PARI) for(n=1, 81, f=factor(n); print1((-1)^sum(s=1, omega(n), f[s, 2]*(Mod(f[s, 1], 4)==3)), ", ")); \\ Arkadiusz Wesolowski, Nov 05 2013

(PARI) a(n)=direuler(p=1, n, if(p==2, 1/(1-kronecker(-4, p)*X)/(1-X), 1/(1-kronecker(-4, p)*X))) /* Ralf Stephan, Mar 27 2015 */

(MAGMA) [KroneckerSymbol(-1, n): n in [1..100]]; // Vincenzo Librandi, Aug 16 2016

CROSSREFS

Cf. A005811, A000364.

The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012

Sequence in context: A108784 A244513 A020985 * A097807 A014077 A174351

Adjacent sequences:  A034944 A034945 A034946 * A034948 A034949 A034950

KEYWORD

sign,nice,easy,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 25 05:43 EST 2020. Contains 338617 sequences. (Running on oeis4.)