A328895 -id:A328895 - 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

A328717 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

Original entry on oeis.org

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

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 = A080891 and s = 2.

			1 - 1/2^2 - 1/3^2 + 1/4^2 + 1/6^2 - 1/7^2 - 1/8^2 + 1/9^2 + ... = 4*Pi^2/(25*sqrt(5)) = 0.70621140325974096993100317576256402766024647185294...

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).
H.-J. Seiffert, Problem B-705, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 4 (1991), p. 372; An Application of a Series Expansion for (arcsinx)^2, Solution to Problem B-705, ibid., Vol. 31, No. 1 (1993), pp. 85-86.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A000045, A001906, A002736, A080891.
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), this sequence (d=5), A328895 (d=8), A258414 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^s: A086466 (s=1), this sequence (s=2), A328723 (s=3).

RealDigits[4*Pi^2/(25*Sqrt[5]), 10, 102] // First

default(realprecision, 100); 4*Pi^2/(25*sqrt(5))

Equals 4*Pi^2/(25*sqrt(5)).
Equals (zeta(2,1/5) - zeta(2,2/5) - zeta(2,3/5) + zeta(2,4/5))/25, where zeta(s,a) is the Hurwitz zeta function.
Equals (polylog(2,u) - polylog(2,u^2) - polylog(2,u^3) + polylog(2,u^4))/sqrt(5), where u = exp(2*Pi*i/5) is a 5th primitive root of unity, i = sqrt(-1).
Equals (polygamma(1,1/5) - polygamma(1,2/5) - polygamma(1,3/5) - polygamma(1,4/5))/25.
Equals Sum_{k>=1} Fibonacci(2*k)/(k^2*binomial(2*k,k)) = Sum_{k>=1} A001906(k)/A002736(k) (Seiffert, 1991). - Amiram Eldar, Jan 17 2022
Equals 1/(Product_{p prime == 1 or 4 (mod 5)} (1 - 1/p^2) * Product_{p prime == 2 or 3 (mod 5)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

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

Original entry on oeis.org

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

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

A346728 Decimal expansion of 11 * Pi^4 / (768 * sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 30 2021

			0.98654286069397050390153449061672691096683375790950...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eqs. (327), (344).

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=4.
Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A328895, A346727, A346260, A346451.

RealDigits[11*Pi^4/(768*Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Equals 11 * Pi^4 / (2^8 * 3 * sqrt(2)).
Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1)^4 + (-1)^k/(4*k+1) ).
Equals Sum_{k>=0} (-1)^floor((k+1)/2) / (2*k+1)^4.

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

A124340 Number of solutions to n = x^2 + 2y^2 + 4(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 7, 8, 10, 8, 12, 16, 8, 16, 18, 14, 18, 16, 16, 20, 24, 16, 21, 24, 20, 32, 28, 16, 32, 32, 20, 36, 32, 28, 36, 36, 24, 32, 42, 32, 42, 40, 28, 48, 48, 32, 57, 42, 36, 48, 52, 40, 40, 64, 36, 56, 58, 32, 60, 64, 56, 64, 48, 40, 66

Offset: 1

Michael Somos, Oct 26 2006

Number 18 of the 74 eta-quotients listed in Table I of Martin (1996).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 4*q^6 + 8*q^7 + 8*q^8 + 7*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A091337, A328895.

Cf. A000122, A000700, A010054, A121373.

with(numtheory):
A091337 := n -> [0, 1, 0, -1, 0, -1, 0, 1][`mod`(n, 8)+1]:
seq(add(A091337(n/d)d, d in divisors(n)), n = 1..60); # Peter Bala, Jan 06 2021

a[n_] := Sum[JacobiSymbol[2, d]*n/d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-François Alcover, Jan 10 2014 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^3 QPochhammer[ q^4] QPochhammer[ q^8]^2 / QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, n / d * kronecker(2, d)))};

{a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 2, p); (p^(e+1) - f^(e+1)) / (p - f)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x + A)^2, n))};

Expansion of q * phi(q) * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^3 * eta(q^4) * eta(q^8)^2 / eta(q)^2 in powers of q.
Euler transform of period 8 sequence [ 2, -1, 2, -2, 2, -1, 2, -4, ...].
a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1, 7 (mod 8), a(p^e) = (p^(e+1) + (-1)^e)/(p + 1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(8*k))^2.
From Peter Bala, Jan 06 2021: (Start)
a(n) = Sum_{ d | n } X(n/d)*d, where X(k) = A091337(k) is a non-principal Dirichlet charcter modulo 8.
G.f.: A(x) = Sum_{n = -oo..oo} (-1)^n*x^(4*n+1)/(1 - x^(4*n+1))^2. (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328895. - Amiram Eldar, Feb 20 2024

A346260 Decimal expansion of 24611 * Pi^8 / (165150720 * sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 30 2021

			0.99984521547922560046287988947718520...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (327).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A328895, A346728, A346727, A346449.

RealDigits[24611 * Pi^8 / (165150720 * Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

Equals 24611 * Pi^8 / (2^19 * 3^2 * 5 * 7 * sqrt(2)).

Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1)^8 + (-1)^k/(4*k+1)^8 ).

A346727 Decimal expansion of 361 * Pi^6 / (245760 * sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 30 2021

Shamos (2011) has incorrect formula 19^2*Pi^4 / (2^14*3*5*sqrt(3)).

			0.998573971953530547670270516106668073031954930...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (327).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers.

Cf. A328895, A346728, A346260, A346450.

RealDigits[361*Pi^6/(245760*Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

Equals 19^2 * Pi^6 / (2^14 * 3 * 5 * sqrt(2)).

Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1)^6 + (-1)^k/(4*k+1)^6 ).

A089003 Number of non-congruent solutions to x^2 - 2y^2 == 1 (mod n).

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 6, 16, 12, 12, 12, 16, 14, 12, 24, 32, 16, 24, 20, 24, 24, 24, 22, 64, 30, 28, 36, 24, 30, 48, 30, 64, 48, 32, 36, 48, 38, 40, 56, 96, 40, 48, 44, 48, 72, 44, 46, 128, 42, 60, 64, 56, 54, 72, 72, 96, 80, 60, 60, 96, 62, 60, 72, 128, 84, 96, 68, 64, 88, 72

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 02 2003

Also, the number of non-congruent solutions to x^2 - 2y^2 == -1 (mod n). - Andrew Howroyd, Jul 16 2018

The comment above is based on the identity -(x^2 - 2y^2) = (x-2y)^2 - 2(x-y)^2. - Jianing Song, Jul 17 2018

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. A088964, A088965, A089002, A062570, A328895.

[n eq 1 select 1 else #[x: x in [1..n], y in [1..n] | (x^2-2*y^2) mod n eq 1]: n in [1..80]]; // Vincenzo Librandi, Jul 16 2018

a[1]=1; a[n_]:=Length@Rest@Union@Flatten@Table[If[Mod[i^2 - 2 j^2, n]==1, i+I j, 0], {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 1, 80}] (* Vincenzo Librandi, Jul 16 2018 *)
f[2, e_] := If[e < 3, 2^e, 2^(e+1)]; f[p_, e_] := If[MemberQ[{1, 7}, Mod[p, 8]], (p - 1), (p + 1)] * p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 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+1)%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(e>2,2,1), p^(e-1)*if(abs(p%8-4)==1, p+1, p-1)))} \\ Andrew Howroyd, Jul 09 2018

Multiplicative with a(2^e) = 2^e for e <= 2, a(2^e) = 2^(e+1) for e > 2, a(p^e) = (p-1)*p^(e-1) for p == +-1 (mod 8), a(p^e) = (p+1)*p^(e-1) for p == +-3 (mod 8). - Andrew Howroyd, Jul 15 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = 9/(16*A328895) = 0.644804064282100795... . - Amiram Eldar, Nov 21 2023

cp's OEIS Frontend

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

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A328717 Decimal expansion of Sum_{k>=1} Kronecker(5,k)/k^2.

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

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

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A346728 Decimal expansion of 11 * Pi^4 / (768 * sqrt(2)).

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

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A124340 Number of solutions to n = x^2 + 2*y^2 + 4*(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.

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A124340 Number of solutions to n = x^2 + 2y^2 + 4(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.