A034947 Jacobi (or Kronecker) symbol (-1/n).

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

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

The congruence in {-1,1} of the odd part of n modulo 4 (Cf. A099545). - Peter Munn, Jul 09 2022

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.

Links

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

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

J.-P. Allouche and Jonathan 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 Jonathan 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), section 38.8.4 Differences of the sum of Gray code digits, coefficients of polynomials L.

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

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 with addendum in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614. a(n) = d(n) at equation 3.1.

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(2*n) = a(n), a(4*n+1) = 1, a(4*n+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)^A065339(n) = lambda(A097706(n)), where A065339(n) is number of primes of the form 4*m + 3 dividing n (counted with multiplicity) and lambda is Liouville's function, A008836. - Arkadiusz Wesolowski, Nov 05 2013 and Peter Munn, Jun 22 2022

Sum_{n>=1} a(n)/n = Pi/2, due to F. von Haeseler; more generally, Sum_{n>=1} a(n)/n^(2*d+1) = Pi^(2*d+1)*A000364(d)/(2^(2*d+2)-2)(2*d)! 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

a(n) = A209615(n) * (-1)^(v2(n)), where v2(n) = A007814(n) is the 2-adic valuation of n. - Jianing Song, Apr 24 2021

a(n) = 2 - A099545(n) == A000265(n) (mod 4). - Peter Munn, Jun 22 2022 and Jul 09 2022

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
(Python)
def A034947(n):
    s = bin(n)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return 1-2*int(s[m-i-2]) if m-i-2 >= 0 else 1 # Chai Wah Wu, Apr 08 2021
(PARI)
a(n) = if(n%2==0, a(n/2), (n+2)%4-2) \\ Peter Munn, Jul 09 2022

Crossrefs

Cf. A000265, A005811, A000364, A008836, A065339, A097706, A099545, A209615.

Moebius transform of A035184.

Cf. A091072 (indices of 1), A091067 (indices of -1), A371594 (indices of run starts).

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: A127252 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