A196525 -id:A196525 - cp's OEIS frontend

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

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

A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

Original entry on oeis.org

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

Offset: 0

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 = A091337 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 + ... = Pi^2/(8*sqrt(2)) = 0.8723580249...

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 at m=8, r=2, s=2.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A091337, A346728, A346727, A346260.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^2, where d is a fundamental discriminant: A309710 (d=-8), A103133 (d=-7), A006752 (d=-4), A086724 (d=-3), A013661 (d=1), A328717 (d=5), this sequence (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), this sequence (s=2), A329715 (s=3).

RealDigits[Pi^2/(8*Sqrt[2]), 10, 102] // First

default(realprecision, 100); Pi^2/(8*sqrt(2))

Equals Pi^2/(8*sqrt(2)).
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 -Integral_{x=0..oo} log(x)/(x^4 + 1) dx. - Amiram Eldar, Jul 17 2020
Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^2) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A118417 a(n) = (2n + 1) 2^(n + 1).

Original entry on oeis.org

2, 12, 40, 112, 288, 704, 1664, 3840, 8704, 19456, 43008, 94208, 204800, 442368, 950272, 2031616, 4325376, 9175040, 19398656, 40894464, 85983232, 180355072, 377487360, 788529152, 1644167168, 3422552064, 7113539584, 14763950080, 30601641984, 63350767616

Offset: 0

Reinhard Zumkeller, Apr 27 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A014480, A118416.

Cf. A002193, A196525, A195695.

[(2*n+1)*2^(n+1): n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

CoefficientList[Series[2 (1 - 3 x^2 + 2 x^3)/((1 - x)^2 (1 - 2 x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 02 2016 *)
Table[(2n+1)2^(n+1),{n,0,30}] (* or *) LinearRecurrence[{4,-4},{2,12},30] (* Harvey P. Dale, Oct 25 2021 *)

a(n) = A118416(n+1,n) = 2*A014480(n).
G.f.: 2*(1-3*x^2+2*x^3)/((1-x)^2*(1-2*x)^2). - Vincenzo Librandi, Sep 02 2016
Sum_{n>=0} 1/a(n) = A196525. - Fred Daniel Kline, May 24 2019
Sum_{n>=0} (-1)^n/a(n) = arctan(1/sqrt(2))/sqrt(2) = A195695 / A002193. - Amiram Eldar, Oct 01 2022

A329715 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^3.

Original entry on oeis.org

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

Offset: 0

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 = A091337 and s = 3.

			1 - 1/3^3 - 1/5^3 + 1/7^3 + 1/9^3 - 1/11^3 - 1/13^3 + 1/15^3 + ... = 0.9583804545...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, Section 2.2 at m=8, r=2, s=3.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A091337.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k^3, where d is a fundamental discriminant: A251809 (d=-8), A327135 (d=-7), A153071 (d=-4), A129404 (d=-3), A002117 (d=1), A328723 (d=5), this sequence (d=8), A329716 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^s: A196525 (s=1), A328895 (s=2), this sequence (s=3).

(PolyGamma[2, 1/8] - PolyGamma[2, 3/8] - PolyGamma[2, 5/8] + PolyGamma[2, 7/8])/(-1024) // RealDigits[#, 10, 102] & // First

Equals (zeta(3,1/8) - zeta(3,3/8) - zeta(3,5/8) + zeta(3,7/8))/512, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(3,u) - polylog(3,u^3) - polylog(3,-u) + polylog(3,-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(2,1/8) - polygamma(2,3/8) - polygamma(2,5/8) + polygamma(2,7/8))/(-1024).
Equals 1/(Product_{p prime == 1 or 7 (mod 8)} (1 - 1/p^3) * Product_{p prime == 3 or 5 (mod 8)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023

A087780 Number of non-congruent solutions to x^2 == 2 mod n.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A038873, A057126, A060594.

Cf. A013661, A196525.

f[2, e_] := Boole[e == 1]; f[p_, e_] := If[MemberQ[{1, 7}, Mod[p, 8]], 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, (f[i,2] == 1), if(f[i,1]%8 == 1 || f[i,1]%8 == 7, 2, 0)));} \\ Amiram Eldar, Nov 21 2023

def A087780(n) :
    res = 1
    for (p, m) in factor(n) :
        if p % 8 in [1, 7] : res *= 2
        elif not (p==2 and m==1) : return 0
    return res
# Eric M. Schmidt, Apr 20 2013

Multiplicative with a(p^m) = 2 for p == 1, 7 (mod 8); a(p^m) = 0 for p == 3, 5 (mod 8); a(2^1) = 1; a(2^m) = 0 for m > 1. - Eric M. Schmidt, Apr 20 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/(sqrt(2)*zeta(2)) = A196525/A013661 = 0.37887551404073012021... . - Amiram Eldar, Nov 21 2023

More terms from David Wasserman, Jun 17 2005

A088964 Number of solutions to x^2 == 2y^2 (mod n).

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 13, 8, 9, 2, 1, 4, 1, 26, 1, 16, 33, 18, 1, 4, 13, 2, 45, 8, 25, 2, 9, 52, 1, 2, 61, 32, 1, 66, 13, 36, 1, 2, 1, 8, 81, 26, 1, 4, 9, 90, 93, 16, 133, 50, 33, 4, 1, 18, 1, 104, 1, 2, 1, 4, 1, 122, 117, 64, 1, 2, 1, 132, 45, 26

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 28 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A087561, A062803, A196525, A328895.

A088964 := proc(n) local a,x,y ; a := 0 ; for x from 0 to n-1 do for y from 0 to n-1 do if (x^2-2*y^2) mod n = 0 then a := a+1 ; end if; end do; end do ; a ; end proc:
seq(A088964(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

a[n_] := Product[{p, e} = pe; Which[p == 2, 2^e, Abs[Mod[p, 8] - 4] == 1, (p^2)^Quotient[e, 2], True, (p+e(p-1))p^(e-1)], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, Apr 08 2020, after Andrew Howroyd *)
f[2, e_] := 2^e; f[p_, e_] := If[MemberQ[{1, 7}, Mod[p, 8]], ((p-1)*e + p)*p^(e-1), p^(2*Floor[e/2])]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n)={my(v=vector(n)); for(i=0, n-1, v[i^2%n + 1]++); sum(i=0, n-1, v[i+1]*v[2*i%n + 1])} \\ Andrew Howroyd, Jul 09 2018

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==2, 2^e, if(abs(p%8-4)==1, (p^2)^(e\2), (p+e*(p-1))*p^(e-1))))} \\ Andrew Howroyd, Jul 09 2018

Multiplicative with a(2^e) = 2^e, a(p^e) = p^(2*floor(e/2)) for p mod 8 = +-3, a(p^e) = ((p-1)*e+p)*p^(e-1) for p mod 8 = +-1. - Andrew Howroyd, Jul 13 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = (64/Pi^4) * A328895 * A196525 = 0.35720726027165235652... . - Amiram Eldar, Nov 21 2023

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

Original entry on oeis.org

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

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 = A175629 and s = 1.

			1 + 1/2 - 1/3 + 1/4 - 1/5 - 1/6 + 1/8 + 1/9 - 1/10 + 1/11 - 1/12 - 1/13 + ... = Pi/sqrt(7) = 1.1874104117...

Sergio A. Carrillo, Where did the examples of Abel's continuity theorem go?, arXiv:2010.10290 [math.HO], 2020. See p. 11.
Eric Weisstein's World of Mathematics, Class Number.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k, where d is a fundamental discriminant: A093954 (d=-8), this sequence (d=-7), A003881 (d=-4), A073010 (d=-3), A086466 (d=5), A196525 (d=8), A196530 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: this sequence (s=1), A103133 (s=2), A327135 (s=3).

RealDigits[Pi/Sqrt[7], 10, 102] // First

PARI
```
default(realprecision, 100); Pi/sqrt(7)
```

Equals Pi/sqrt(7). This is related to the class number formula: if d<0 is the fundamental discriminant of an imaginary quadratic number field, Chi(k) = Kronecker(d,k), then L(1,Chi) = Sum_{k>=1} Kronecker(d,k)/k = 2*Pi*h(d)/(sqrt(|d|)*w(d)), where h(d) is the class number of K = Q[sqrt(d)], w(d) is the number of elements in K whose norms are 1 (w(d) = 6 if d = -3, 4 if d = -4 and 2 if d < -4). Here d = -7, h(d) = 1, w(d) = 2.
Equals (polylog(1,u) + polylog(1,u^2) - polylog(1,u^3) + polylog(1,u^4) - polylog(1,u^5) - polylog(1,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
Equals (polygamma(0,1/7) + polygamma(0,2/7) - polygamma(0,3/7) + polygamma(0,4/7) - polygamma(0,5/7) - polygamma(0,6/7))/49.
Equals 1/Product_{p prime} (1 - Kronecker(-7,p)/p), where Kronecker(-7,p) = 0 if p = 7, 1 if p == 1, 2 or 4 (mod 7) or -1 if p == 3, 5 or 6 (mod 7). - Amiram Eldar, Dec 17 2023

A346781 a(n) is the numerator of the sum of the first n terms of 1 - 1/3 - 1/5 + 1/7 + 1/9 - 1/11 - 1/13 + ... .

Original entry on oeis.org

1, 2, 7, 64, 227, 2182, 24901, 27904, 519413, 9103082, 8410247, 207985216, 1106853941, 3134651098, 85885292267, 2808012157952, 2944757946677, 402260886146, 14238994069127, 14850593365952, 632726700580207, 26229300849325726, 25294817897063581, 1230908174013784832

Offset: 1

Hugo Pfoertner, Aug 03 2021

The limit for n->oo of the sum 1 - 1/3 - 1/5 + 1/7 + ... is log(1+sqrt(2))/sqrt(2) (A196525). See there for more information.

			1, 2/3, 7/15, 64/105, 227/315, 2182/3465, 24901/45045, 27904/45045, ...

Barry Mazur, Chapter IV.1 Algebraic Numbers, page 316, in The Princeton Companion to Mathematics, ed. Timothy Gowers, Princeton University Press, Princeton and Oxford, 2008.

The corresponding denominators are A025547.

Cf. A091337, A196525.

a346781(limit)={my(s=0,b(n)=1/(n*sign(4-(n+2)%8)));forstep(k=1,limit,2,print1(numerator(s+=b(k)),", "))};
a346781(47)

cp's OEIS Frontend

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

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Extensions

A328895 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^2.

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Mathematica

PARI

Formula

A118417 a(n) = (2*n + 1) * 2^(n + 1).

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Magma

Mathematica

Formula

A329715 Decimal expansion of Sum_{k>=1} Kronecker(8,k)/k^3.

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Mathematica

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A087780 Number of non-congruent solutions to x^2 == 2 mod n.

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Mathematica

PARI

Sage

Formula

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A088964 Number of solutions to x^2 == 2y^2 (mod n).

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Maple

Mathematica

PARI

PARI

Formula

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

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A118417 a(n) = (2n + 1) 2^(n + 1).