A188510 -id:A188510 - cp's OEIS frontend

A309710 Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^2.

1, 0, 6, 4, 7, 3, 4, 1, 7, 1, 0, 4, 3, 5, 0, 3, 3, 7, 0, 3, 9, 2, 8, 2, 7, 4, 5, 1, 4, 6, 1, 6, 6, 8, 8, 8, 9, 4, 8, 3, 0, 9, 9, 1, 5, 1, 7, 7, 4, 4, 8, 5, 1, 2, 4, 4, 1, 9, 8, 7, 4, 5, 0, 8, 0, 6, 3, 9, 9, 0, 1, 7, 1, 7, 5, 8, 6, 4, 3, 7, 6, 3, 6, 6, 6, 5, 3, 4, 2, 5, 0

Offset: 1

Jianing Song, Nov 19 2019

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A188510 and s = 2.

			1 + 1/3^2 - 1/5^2 - 1/7^2 + 1/9^2 + 1/11^2 - 1/13^2 - 1/15^2 + ...= 1.0647341710...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 L(m=8, r=4, s=2).
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A188510.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: this sequence (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), A328717 (d=5), A328895 (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^s: A093954 (s=1), this sequence (s=2), A251809 (s=3).

(PolyGamma[1, 1/8] + PolyGamma[1, 3/8] - PolyGamma[1, 5/8] - PolyGamma[1, 7/8])/64 // RealDigits[#, 10, 102] & // First

Equals (zeta(2,1/8) + zeta(2,3/8) - zeta(2,5/8) - zeta(2,7/8))/64, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(2,u) + polylog(2,u^3) - polylog(2,-u) - polylog(2,-u^3))/sqrt(-8), where u = sqrt(2)/2 + i*sqrt(2)/2 is an 8th primitive root of unity, i = sqrt(-1).
Equals (polygamma(1,1/8) + polygamma(1,3/8) - polygamma(1,5/8) - polygamma(1,7/8))/64.
Equals 1/(Product_{p prime == 1 or 3 (mod 8)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 8)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

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)

A257170 Expansion of (1 + x) * (1 + x^3) / (1 + x^4) in powers of x.

Original entry on oeis.org

1, 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

Michael Somos, Apr 17 2015

			G.f. = 1 + x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, -1).

Cf. A188510.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+x^3)/(1+x^4))); // G. C. Greubel, Aug 02 2018

a[ n_] := If[ EvenQ[ n], Boole[n == 0], (-1)^Quotient[ n, 4]];
a[ n_] := If[ n == 0, 1, Sign[ n] SeriesCoefficient[ (1 + x) * (1 + x^3) / (1 + x^4), {x, 0, Abs @ n}]];
CoefficientList[Series[(1+x)*(1+x^3)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)

{a(n) = if( n%2 == 0, n==0, (-1)^(n\4))};

{a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x) * (1 + x^3) / (1 + x^4), + x* O(x^abs(n)), abs(n)))};

x='x+O('x^60); Vec((1+x)*(1+x^3)/(1+x^4)) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 8 sequence [1, -1, 1, -1, 0, -1, 0, 1].
a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e otherwise and a(0) = 1.
a(n) = -a(-n) for all n in Z unless n = 0. a(n+4) = -a(n) unless n = 0 or n = -4. a(2*n) = 0 unless n = 0.
a(n) = A188510(n) unless n = 0.
a(n+1) - a(n) = (-1)^n if n>0.
G.f.: (1 + x) * (1 + x^3) / (1 + x^4) = 1 + (x + x^3) / (1 + x^4).
G.f.: (1 - x^2) * (1 - x^4) * (1 - x^6) / ((1 - x) * (1 - x^3) * (1 - x^8)).
G.f.: 1 / (1 - x / (1 + x / (1 + x / (1 - x / (1 + 2*x / (1 - 2*x / (1 - x / (2 + x)))))))).

A257477 Multiplicative with a(2) = 0, a(4) = -1, a(2^e) = 0 if e>2, a(3) = -1, a(3^e) = 0^e if e>1, a(p^e) = 1 if p == 1, 3 (mod 8), a(p^e) = (-1)^e if p == 5, 7 (mod 8).

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 25 2015

			G.f. = x - x^3 - x^4 - x^5 - x^7 + x^11 + x^12 - x^13 + x^15 + x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A168182, A188510, A257403.

a[ n_] := Sign[n] If[ Abs[n] < 2, 1, Times @@ (Which[ # < 5, -Boole[# + #2 == 4], Mod[#, 8] < 4, 1, True, (-1)^#2] & @@@ FactorInteger[Abs@n])];
f[x_] := (x + x^3)/(1 + x^4); CoefficientList[Series[f[x] - 2*f[x^3] - f[x^4] + f[x^9] + 2*f[x^12] - f[x^36], {x,0,50}], x] (* G. C. Greubel, Aug 03 2018 *)

{a(n) = my(A, p, e); if( n==0, 0, A = factor(abs(n)); sign(n) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, -(p+e==4), if( p%8 < 4, 1, (-1)^e))))};

G.f.: f(x) - 2*f(x^3) - f(x^4) + f(x^9) + 2*f(x^12) - f(x^36) where f(x) = (x + x^3) / (1 + x^4) is the g.f. for A188510.
abs(a(2*n + 1)) = A168182(n+5).
a(4*n + 2) = a(8*n) = a(9*n) = 0.
a(n) = -a(-n) = a(n + 288) for all n in Z.
Moebius transform of A257403.
Sum_{k=1..n} abs(a(k)) ~ 5*n/9. - Amiram Eldar, Jan 29 2024

Definition corrected by Georg Fischer, Jul 23 2022

A315520 Coordination sequence Gal.4.140.3 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

Original entry on oeis.org

1, 6, 11, 17, 22, 27, 33, 38, 44, 50, 55, 61, 66, 71, 77, 82, 88, 94, 99, 105, 110, 115, 121, 126, 132, 138, 143, 149, 154, 159, 165, 170, 176, 182, 187, 193, 198, 203, 209, 214, 220, 226, 231, 237, 242, 247, 253, 258, 264, 270

Offset: 0

Brian Galebach and N. J. A. Sloane, Jun 18 2018

nonn

Note that there may be other vertices in the Galebach list of u-uniform tilings with u <= 6 that have this same coordination sequence. See the Galebach link for the complete list of A-numbers for all these tilings.

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

Cf. A188510, A250120.

Conjectures from Chai Wah Wu, Jun 11 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) - a(n-4) + 2*a(n-5) - a(n-6) for n > 6.
G.f.: (x^6 + 4*x^5 + x^3 + 4*x + 1)/((x - 1)^2*(x^4 + 1)). (End)
Conjecture: a(n) = (22*n + 2*A188510(n))/4 for n > 0. - Stefano Spezia, Jul 30 2022

A323378 Square array read by antidiagonals: T(n,k) = Kronecker symbol (-n/k), n >= 1, k >= 1.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Jan 12 2019

If A215200 is arranged into a square array A215200(n,k) = kronecker symbol(n/k) with n >= 0, k >= 1, then this sequence gives the other half of the array.

Note that there is no such n such that the n-th row and the n-th column are the same.

			Table begins
  1,  1, -1,  1,  1, -1, -1,  1,  1,  1, ... ((-1/k) = A034947)
  1,  0,  1,  0, -1,  0, -1,  0,  1,  0, ... ((-2/k) = A188510)
  1, -1,  0,  1, -1,  0,  1, -1,  0,  1, ... ((-3/k) = A102283)
  1,  0, -1,  0,  1,  0, -1,  0,  1,  0, ... ((-4/k) = A101455)
  1, -1,  1,  1,  0, -1,  1, -1,  1,  0, ... ((-5/k) = A226162)
  1,  0,  0,  0,  1,  0,  1,  0,  0,  0, ... ((-6/k) = A109017)
  1,  1, -1,  1, -1, -1,  0,  1,  1, -1, ... ((-7/k) = A175629)
  1,  0,  1,  0, -1,  0, -1,  0,  1,  0, ... ((-8/k) = A188510)
  ...

Eric Weisstein's World of Mathematics, Kronecker Symbol.

Cf. A215200.

The first rows are listed in A034947, A188510, A102283, A101455, A226162, A109017, A175629, A188510, ...

PARI
```
T(n,k) = kronecker(-n, k)
```

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A309710 Decimal expansion of Sum_{k>=1} Kronecker(-8,k)/k^2.

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

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A257170 Expansion of (1 + x) * (1 + x^3) / (1 + x^4) in powers of x.

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A257477 Multiplicative with a(2) = 0, a(4) = -1, a(2^e) = 0 if e>2, a(3) = -1, a(3^e) = 0^e if e>1, a(p^e) = 1 if p == 1, 3 (mod 8), a(p^e) = (-1)^e if p == 5, 7 (mod 8).

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A315520 Coordination sequence Gal.4.140.3 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

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A323378 Square array read by antidiagonals: T(n,k) = Kronecker symbol (-n/k), n >= 1, k >= 1.

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