A011585 -id:A011585 - cp's OEIS frontend

A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0

Offset: 0

Gerald McGarvey, Jan 20 2005

Called X(n) (i.e., Chi(n)) in Hardy and Wright (p. 241), who show that X(n*m) = X(n)*X(m) for all n and m (i.e., X(n) is completely multiplicative) since (n*m - 1)/2 - (n - 1)/2 - (m - 1)/2 = (n - 1)*(m - 1)/2 == 0 (mod 2) when n and m are odd.
Same as A056594 but with offset 1.
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the non-principal Dirichlet character mod 4. (The principal character is A000035.)
Associated Dirichlet L-functions are for example L(1,chi) = Sum_{n>=1} a(n)/n = A003881, or L(2,chi) = Sum_{n>=1} a(n)/n^2 = A006752, or L(3,chi) = Sum_{n>=1} a(n)/n^3 = A153071. (End)
a(n) is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 0, y = -1, z is arbitrary. - Michael Somos, Nov 27 2019

			G.f. = x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + x^21 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=4, Chi_2(n).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1979, p. 241.

Muniru A Asiru, Table of n, a(n) for n = 0..1000 (a(0) prepended by Jianing Song)
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Clark Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17.
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A002654, A009116, A009454, A056594, A108520, A111335, A126120, A146559.

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), this sequence (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).

a := [1, 0];; for n in [3..10^2] do a[n] := a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018

m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/(1+x^2))); // G. C. Greubel, Aug 23 2018

a := n -> `if`(n mod 2=0, 0, (-1)^((n-1)/2)):
seq(a(n), n=1..10^3); # Muniru A Asiru, Feb 02 2018

a[ n_] := {1, 0, -1, 0}[[ Mod[ n, 4, 1]]]; (* Michael Somos, Jan 13 2014 *)
LinearRecurrence[{0, -1}, {1, 0}, 75] (* G. C. Greubel, Aug 23 2018 *)

{a(n) = if( n%2, (-1)^(n\2))}; /* Michael Somos, Sep 02 2005 */

{a(n) = kronecker( -4, n)}; /* Michael Somos, Mar 30 2012 */

def A101455(n): return (0,1,0,-1)[n&3] # Chai Wah Wu, Jun 21 2024

Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. - Mitch Harris May 17 2005
Euler transform of length 4 sequence [0, -1, 0, 1]. - Michael Somos, Sep 02 2005
G.f.: (x - x^3)/(1 - x^4) = x/(1 + x^2). - Michael Somos, Sep 02 2005
G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 * (1 + 2*v). - Michael Somos, Aug 04 2011
a(n + 4) = a(n), a(n + 2) = a(-n) = -a(n), a(2*n) = 0, a(2*n + 1) = (-1)^n for all n in Z. - Michael Somos, Aug 04 2011
a(n + 1) = A056594(n). - Michael Somos, Jan 13 2014
REVERT transform is A126120. STIRLING transform of A009454. BINOMIAL transform is A146559. BINOMIAL transform of A009116. BIN1 transform is A108520. MOBIUS transform of A002654. EULER transform is A111335. - Michael Somos, Mar 30 2012
Completely multiplicative with a(p) = 2 - (p mod 4). - Werner Schulte, Feb 01 2018
a(n) = (-(n mod 2))^binomial(n, 2). - Peter Luschny, Sep 08 2018
a(n) = sin(n*Pi/2) = Im(i^n) where i is the imaginary unit. - Jianing Song, Sep 09 2018
From Jianing Song, Nov 14 2018: (Start)
a(n) = ((-4)/n) (or more generally, ((-4^i)/n) for i > 0), where (k/n) is the Kronecker symbol.
E.g.f.: sin(x).
Dirichlet g.f. is the Dirichlet beta function.
a(n) = A091337(n)*A188510(n). (End)

a(0) prepended by Jianing Song, Nov 14 2024

A091337 a(n) = (2/n), where (k/n) is the Kronecker symbol.

Original entry on oeis.org

0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1

Offset: 0

Eric W. Weisstein, Dec 30 2003

Sinh(1) in 'reflected factorial' base is 1.01010101010101010101010101010101010101010101... see A073097 for cosh(1). - Robert G. Wilson v, May 04 2005
A non-principal character for the Dirichlet L-series modulo 8, see arXiv:1008.2547 and L-values Sum_{n >= 1} a(n)/n^s in eq (318) by Jolley. - R. J. Mathar, Oct 06 2011 [The other two non-principal characters are A101455 = {(-4/n)} and A188510 = {(-2/n)}. - Jianing Song, Nov 14 2024]
Period 8: repeat [0, 1, 0, -1, 0, -1, 0, 1]. - Wesley Ivan Hurt, Sep 07 2015 [Adapted by Jianing Song, Nov 14 2024 to include a(0) = 0.]
a(n) = (2^(2i+1)/n), where (k/n) is the Kronecker symbol and i >= 0. - A.H.M. Smeets, Jan 23 2018

			G.f. = x - x^3 - x^5 + x^7 + x^9 - x^11 - x^13 + x^15 + x^17 - x^19 - x^21 + ...

L. B. W. Jolley, Summation of series, Dover (1961).

Jianing Song, Table of n, a(n) for n = 0..1000
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.
R. J. Mathar, Table of Dirichlet L-series..., arXiv:1008.2547 [math.NT], 2010, 2015, L(m=8,r=2,s).
Michael Somos, Rational Function Multiplicative Coefficients
Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A000129, A035185, A073097, A087960, A102587.

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), this sequence (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).

[(n mod 2) * (-1)^((n+1) div 4)  : n in [1..100]]; // Vincenzo Librandi, Oct 31 2014

A091337:= n -> [0, 1, 0, -1, 0, -1, 0, 1][(n mod 8)+1]: seq(A091337(n), n=1..100); # Wesley Ivan Hurt, Sep 07 2015

KroneckerSymbol[Range[100], 2] (* Alonso del Arte, Oct 30 2014 *)

{a(n) = (n%2) * (-1)^((n+1)\4)}; /* Michael Somos, Sep 10 2005 */

{a(n) = kronecker( 2, n)}; /* Michael Somos, Sep 10 2005 */

{a(n) = [0, 1, 0, -1, 0, -1, 0, 1][n%8 + 1]}; /* Michael Somos, Jul 17 2009 */

Euler transform of length 8 sequence [0, -1, 0, -1, 0, 0, 0, 1]. - Michael Somos, Jul 17 2009
a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1, 7 (mod 8), a(p^e) = (-1)^e if p == 3, 5 (mod 8). - Michael Somos, Jul 17 2009
G.f.: x*(1 - x^2)/(1 + x^4). a(n) = -a(n + 4) = a(-n) for all n in Z. a(2*n) = 0. a(2*n + 1) = A087960(n). - Michael Somos, Apr 10 2011
Transform of Pell numbers A000129 by the Riordan array A102587. - Paul Barry, Jul 14 2005
a(n) = (2/n) = (n/2), Charles R Greathouse IV explained. - Alonso del Arte, Oct 31 2014
a(n) = (1 - (-1)^n)*(-1)^(n/4 - 1/8 - (-1)^n/8 + (-1)^((2*n + 1 - (-1)^n)/4)/4)/2. - Wesley Ivan Hurt, Sep 07 2015
From Jianing Song, Nov 14 2018: (Start)
a(n) = sqrt(2)*sin(Pi*n/2)*sin(Pi*n/4).
E.g.f.: sqrt(2)*cos(x/sqrt(2))*sinh(x/sqrt(2)).
Moebius transform of A035185.
a(n) = A101455(n)*A188510(n). (End)
a(n) = Sum_{i=1..n} (-1)^(i + floor((i-3)/4)). - Wesley Ivan Hurt, Apr 27 2020
Sum_{n>=1} a(n)/n = A196525. Sum_{n>=1} a(n)/n^2 = A328895. Sum_{n>=1} a(n)/n^3 = A329715. Sum_{n>=1} a(n)/n^4 = A346728. - R. J. Mathar, Dec 17 2024

a(0) prepended by Jianing Song, Nov 14 2024

A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -19.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 1, 2, 0, 1, 2, 0, 0, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 1

Offset: 1

N. J. A. Sloane

From Jianing Song, Sep 07 2018: (Start)
Half of the number of integer solutions to x^2 + x*y + 5*y^2 = n.
Inverse Moebius transform of A011585. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -19. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A028641.
Moebius transform gives A011585.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-19, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

m = -19; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

From Jianing Song, Sep 07 2018: (Start)
a(n) is multiplicative with a(19^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-19, p) = -1, a(p^e) = e + 1 if Kronecker(-19, p) = 1.
G.f.: Sum_{k>0} Kronecker(-19, k) * x^k / (1 - x^k).
A028641(n) = 2 * a(n) unless n = 0.
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(19) = 0.720730... . - Amiram Eldar, Oct 11 2022

A289741 a(n) = Kronecker symbol (-20/n).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0

Offset: 0

Jianing Song, Dec 27 2018

Period 20: repeat [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1].
This sequence is one of the three non-principal real Dirichlet characters modulo 20. The other two are Jacobi or Kronecker symbols {(20/n)} (or {(n/20)}) and {((-100)/n)} (A185276).
Note that (Sum_{i=0..19} i*a(i))/(-20) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-5)). The fact Q(sqrt(-5)) has class number 2 implies that Q(sqrt(-5)) is not a unique factorization domain.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,-1).

Cf. A035170 (inverse Moebius transform).
Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), this sequence (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).
Cf. A045797, A045798, A185276.

Array[KroneckerSymbol[-20, #]&, 100, 0] (* Amiram Eldar, Jan 10 2019 *)

PARI
```
a(n) = kronecker(-20, n)
```

a(n) = 1 for n in A045797; -1 for n in A045798; 0 for n that are not coprime with 20.
Completely multiplicative with a(p) = a(p mod 20) for primes p.
a(n) = A080891(n)*A101455(n).
a(n) = -a(n+10) = -a(-n) for all n in Z.
Multiplicative with a(2) = a(5) = 0, a(p) = (-1)^floor(p/10) otherwise; equivalently: a(n) = (-1)^floor(n/10) if n is coprime to 2*5, 0 otherwise. - M. F. Hasler, Feb 28 2022

A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

Original entry on oeis.org

4, 5, 7, 9, 11, 17, 19, 23, 43, 47, 61, 73, 83, 101, 131, 137, 139, 149, 157, 163, 169, 191, 197, 199, 229, 233, 239, 251, 263, 271, 277, 283, 311, 313, 347, 349, 353, 359, 367, 389, 397, 419, 443, 457, 461, 463, 467, 479, 491, 499, 503, 541, 557, 571

Offset: 1

Jianing Song, Feb 19 2021

Also norms of prime ideals in Z[(1+sqrt(-19))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
Consists of the primes such that (p,19) >= 0 and the squares of primes such that (p,19) = -1, where (p,19) is the Legendre symbol.
For primes p such that (p,19) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-19))/2], namely (x + y*(1+sqrt(-19))/2) and (x + y*(1-sqrt(-19))/2), where (x,y) is a solution to x^2 + x*y + 5*y^2 = p; for p = 19, (sqrt(-19)) is the unique ideal with norm p; for primes p with (p,19) = -1, (p) is the only ideal with norm p^2.

			norm((1 + sqrt(-19))/2) = norm((1 - sqrt(-19))/2) = 5;
norm((3 + sqrt(-19))/2) = norm((3 - sqrt(-19))/2) = 7;
norm((5 + sqrt(-19))/2) = norm((5 - sqrt(-19))/2) = 11;
norm((7 + sqrt(-19))/2) = norm((7 - sqrt(-19))/2) = 17.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A011585, A106863, A191019, A191063.
The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035171.
The total number of elements with norm n is given by A028641.
Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), this sequence (D=-19), A091727 (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

isA341787(n) = my(disc=-19); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

A322796 a(n) = Kronecker symbol (6/n).

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0

Offset: 0

Jianing Song, Dec 26 2018

Period 24: repeat [0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1].
Also a(n) = Kronecker symbol (24/n).
This sequence is one of the seven non-principal real Dirichlet characters modulo 24. The other six are Jacobi or Kronecker symbols {(-6/n)} (or {(n/6)}, {(-24/n)}, {(n/24)}, A109017), {(-12/n)} (or {(n/12)}, A134667), {(12/n)} (A110161), {(-18/n)} (or {(-72/n)}), {(18/n)} (or {(72/n)}, {(n/72)}) and {(-36/n)}. These sequences all become the same after taking absolute values.

Antti Karttunen, Table of n, a(n) for n = 0..65543
Eric Weisstein's World of Mathematics, Kronecker Symbol (contains this sequence)
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,-1).

Cf. A035188 (inverse Moebius transform).

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), this sequence (d=24).

[KroneckerSymbol(6, n): n in [0..100]]; // Vincenzo Librandi, Jan 01 2019

Array[KroneckerSymbol[6, #] &, 105, 0] (* Michael De Vlieger, Dec 31 2018 *)
Table[KroneckerSymbol[6, n], {n, 0, 100}] (* Vincenzo Librandi, Jan 01 2019 *)

a(n) = kronecker(6, n); \\ --- Argument order corrected by Antti Karttunen, Sep 27 2019

a(n) = 1 for n == 1, 5, 19, 23 (mod 24); -1 for n == 7, 11, 13, 17 (mod 24); 0 for n that are not coprime with 21.
Completely multiplicative with a(p) = a(p mod 24) for primes p.
a(n) = A091337(n)*A102283(n).
a(n) = A109017(n+12) = A109017(n-12).
a(n) = a(-n) = a(n+24) for all n in Z.

Definition corrected by Antti Karttunen, Sep 28 2019

A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

Original entry on oeis.org

0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1, 0

Offset: 0

Jianing Song, Dec 27 2018

Period 21: repeat [0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1].
Also a(n) = Kronecker symbol (21/n).
This sequence is one of the three non-principal real Dirichlet characters modulo 21. The other two are Jacobi or Kronecker symbols {(n/63)} (or {(-63/n)}) and {(n/147)} (or {(-147/n)}).

Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,0,-1,0,1,-1,0,1,-1).

Cf. A035203 (inverse Moebius transform).

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017(d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), this sequence (d=21), A322796 (d=24).

JacobiSymbol[Range[0, 100], 21] (* Paolo Xausa, Mar 19 2025 *)

PARI
```
a(n) = kronecker(n, 21)
```

a(n) = 1 for n == 1, 4, 5, 16, 17, 20 (mod 21); -1 for n == 2, 8, 10, 11, 13, 19 (mod 21); 0 for n that are not coprime with 21.
Completely multiplicative with a(p) = a(p mod 21) for primes p.
a(n) = A102283(n)*A175629(n).
a(n) = a(n+21) = -a(n) for all n in Z.
From Chai Wah Wu, Feb 18 2021: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-8) - a(n-9) + a(n-11) - a(n-12) for n > 11.
G.f.: -x*(x - 1)*(x + 1)*(x^8 - 2*x^7 + 2*x^6 + 2*x^2 - 2*x + 1)/(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1). (End)

cp's OEIS Frontend

A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...

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A091337 a(n) = (2/n), where (k/n) is the Kronecker symbol.

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A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -19.

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A289741 a(n) = Kronecker symbol (-20/n).

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A341787 Norms of prime elements in Z[(1+sqrt(-19))/2], the ring of integers of Q(sqrt(-19)).

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A322796 a(n) = Kronecker symbol (6/n).

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A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -19.