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

Author

N. J. A. Sloane

Keywords

• sign
• nice
• easy
• mult

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

Examples

			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 + ...
		

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 Jeffrey 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.
• L. Almodovar, V. H. Moll, H. Quand, Infinite products arising in paperfolding, JIS 19 (2016) # 16.5.1 eq. (1)
• 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.
• Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
• A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
• Jeffrey Shallit, Cloitre's Self-Generating Sequence, arXiv:2501.00784 [math.CO], 2025. See page 4.
• Eric Weisstein's World of Mathematics, Kronecker Symbol.
• Index entries for sequences obtained by enumerating foldings.

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

Programs

Magma

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

Maple

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

Mathematica

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

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 */

PARI

a(n) = if(n%2==0, a(n/2), (n+2)%4-2) \\ Peter Munn, Jul 09 2022

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

Python

def A034947(n): return -1 if n>>(-n&n).bit_length()&1 else 1 # Chai Wah Wu, Feb 26 2025

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