A259171 -id:A259171 - cp's OEIS frontend

A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

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

Offset: 1

Benoit Cloitre, Jan 07 2006

Known to be transcendental. After n=2 it is the same as A100554(n).

Imaginary part of psi(I) (for the real part, see A248177). - Stanislav Sykora, Oct 03 2014

			2.076674047468581174134050794750000490445656266403816665575062484390...

Michel Waldschmidt, Elliptic functions and transcendance, Surveys in number theory, 143-188, Dev. Math., 17, Springer, New York, 2008.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Digamma function.
Index entries for transcendental numbers.

Cf. A013661 (Sum_{i>=1} 1/i^2), A232883 (Sum_{i>=0} 1/(2*i^2+1)). - Bruno Berselli, Dec 02 2013
Cf. A248177.
Essentially the same as A100554 and A259171.

RealDigits[N[Im[PolyGamma[0,I]],105]][[1]]  (* Vaclav Kotesovec, Oct 03 2014 *)

PARI
```
1/2+Pi/tanh(Pi)/2
```

imag(psi(I)) \\ Stanislav Sykora, Oct 03 2014

sumnumrat(1/(x^2+1),0) \\ Charles R Greathouse IV, Jan 20 2022

Equals 1/2 + Pi /(2*tanh(Pi)).
Equals 1+Integral_{x >= 0} sin(x)/(exp(x)-1) dx. - Robert FERREOL, Jan 12 2016.
Equals Sum_{k>=0} (-1)^(k+1)*(zeta(2*k) - 1). - Amiram Eldar, Apr 28 2025

Offset changed from 0 to 1 by Bruno Berselli, Dec 02 2013

A071253 a(n) = n^2*(n^2+1).

Original entry on oeis.org

0, 2, 20, 90, 272, 650, 1332, 2450, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 105300, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1049600, 1187010, 1337492, 1501850, 1680912

Offset: 0

N. J. A. Sloane, Jun 12 2002

The identity (n^5 + n^3)^2 + (n^2*(n^2 + 1))^2 = n*(n^3 + n)^3 can be written as A155977(n)^2 + a(n)^2 = n*A034262(n)^3. - Vincenzo Librandi, Aug 08 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A002522, A002378, A034262, A037270, A069187, A155977.

A071253:= func< n | 2*Binomial(n^2+1,2) >;
[A071253(n): n in [0..40]]; // G. C. Greubel, Sep 12 2024

with(combinat):seq(lcm(fibonacci(3,n),n^2),n=0..35); # Zerinvary Lajos, Apr 20 2008
a:=n->add(n+add(n+add(n, j=1..n-1),j=1..n),j=1..n):seq(a(n), n=0..41); # Zerinvary Lajos, Aug 27 2008

Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n,0,40}] (* Artur Jasinski, Feb 10 2010 *)
Table[n^2*(n^2+1),{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[2 x (1+x) (1+4 x+x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015

def A071253(n): return 2*binomial(n^2+1,2)
[A071253(n) for n in range(41)] # G. C. Greubel, Sep 12 2024

a(n) = A002522(n)*A000290(n). - Zerinvary Lajos, Apr 20 2008
a(n) = (1/4)*sinh(2*arcsinh(n))^2. - Artur Jasinski, Feb 10 2010
G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Jan 08 2012
a(n) = A002378(A000290(n)). - Rick L. Shepherd, Sep 22 2014
Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Oct 18 2019
a(n) = 2*A037270(n). - R. J. Mathar, Oct 18 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - Amiram Eldar, Nov 05 2020
E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - Stefano Spezia, Oct 08 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 16 2023

A329083 Decimal expansion of Sum_{k>=0} 1/(k^2+2).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(2).
This and A329090 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329090.

			1.36102810057372792282...

Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), this sequence (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 + Sqrt[2]*Pi*Coth[Sqrt[2]*Pi])/4, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(2)

sumnumrat(1/(x^2+2),0) \\ Charles R Greathouse IV, Jan 20 2022

Equals (1 + (sqrt(2)*Pi)*coth(sqrt(2)*Pi))/4 = (1 + (sqrt(-2)*Pi)*cot(sqrt(-2)*Pi))/4.

A329084 Decimal expansion of Sum_{k>=0} 1/(k^2+3).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(3).

			1.07360040991518411591...

Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), this sequence (F(3)), A329085 (F(4)), A329086 (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[Sum[1/(k^2+3),{k,0,\[Infinity]}],10,120][[1]] (* Harvey P. Dale, Jul 05 2021 *)
RealDigits[(1 + Sqrt[3]*Pi*Coth[Sqrt[3]*Pi])/6, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(3)

sumnumrat(1/(x^2+3), 0) \\ Charles R Greathouse IV, Jan 20 2022

Equals (1 + (sqrt(3)*Pi)*coth(sqrt(3)*Pi))/6 = (1 + (sqrt(-3)*Pi)*cot(sqrt(-3)*Pi))/6.

A329085 Decimal expansion of Sum_{k>=0} 1/(k^2+4).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(4).
This and A329092 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329092.

			0.91040364132111511419...

Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), this sequence (F(4)), A329086 (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 + 2*Pi*Coth[2*Pi])/8, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(4)

sumnumrat(1/(x^2+4), 0) \\ Charles R Greathouse IV, Jan 20 2022

Equals (1 + (2*Pi)*coth(2*Pi))/8 = (1 + (2*Pi*i)*cot(2*Pi*i))/8, i = sqrt(-1).

A329080 Decimal expansion of Sum_{k>=0} 1/(k^2-5), negated.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(-5) (negated).
This and A329087 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329087.

			-0.86683259566274485298...

Cf. this sequence (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 + Sqrt[5]*Pi*Cot[Sqrt[5]*Pi])/10, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(-5)

sumnumrat(1/(x^2-5), 0) \\ Charles R Greathouse IV, Jan 20 2022

Equals (1 + (sqrt(-5)*Pi)*coth(sqrt(-5)*Pi))/(-10).

Equals (1 + (sqrt(5)*Pi)*cot(sqrt(5)*Pi))/(-10).

A329081 Decimal expansion of Sum_{k>=0} 1/(k^2-3).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(-3).

			0.64331685665276028377...

Cf. A329080 (F(-5)), this sequence (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 + Sqrt[3]*Pi*Cot[Sqrt[3]*Pi])/6, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(-3)

sumnumrat(1/(x^2-3), 0) \\ Charles R Greathouse IV, Jan 20 2022

Equals (1 + (sqrt(-3)*Pi)*coth(sqrt(-3)*Pi))/(-6) = (1 + (sqrt(3)*Pi)*cot(sqrt(3)*Pi))/(-6).

A329082 Decimal expansion of Sum_{k>=0} 1/(k^2-2), negated.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(-2) (negated).
This and A329089 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329089.

			-0.55681040770062008255...

Cf. A329080 (F(-5)), A329081 (F(-3)), this sequence (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 + Sqrt[2]*Pi*Cot[Sqrt[2]*Pi])/4, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(-2)

sumnumrat(1/(x^2-2), 0) \\ Charles R Greathouse IV, Jan 20 2022

Equals (1 + (sqrt(-2)*Pi)*coth(sqrt(-2)*Pi))/(-4) = (1 + (sqrt(2)*Pi)*cot(sqrt(2)*Pi))/(-4).

A329086 Decimal expansion of Sum_{k>=0} 1/(k^2+5).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives F(5).
This and A329093 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329093.

			Sum_{k>=0} 1/(k^2+5) = 0.80248258480678688683...

Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), this sequence (F(5)).

Cf. A329087 (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 + Sqrt[5]*Pi*Coth[Sqrt[5]*Pi])/10, 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

default(realprecision, 100); my(F(x) = (1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); F(5)

sumnumrat(1/(x^2+5), 0) \\ Charles R Greathouse IV, Jan 20 2022

Sum_{k>=0} 1/(k^2+5) = (1 + (sqrt(5)*Pi)*coth(sqrt(5)*Pi))/10 = (1 + (sqrt(-5)*Pi)*cot(sqrt(-5)*Pi))/10.

A329087 Decimal expansion of Sum_{k>=1} 1/(k^2-5), negated.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 04 2019

In general, for complex numbers z, if we define F(z) = Sum_{k>=0} 1/(k^2+z), f(z) = Sum_{k>=1} 1/(k^2+z), then we have:
F(z) = (1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...;
f(z) = (-1 + sqrt(z)*Pi*coth(sqrt(z)*Pi))/(2z), z != 0, -1, -4, -9, -16, ...; Pi^2/6, z = 0. Note that f(z) is continuous at z = 0.
This sequence gives f(-5) (negated).
This and A329080 are essentially the same, but both sequences are added because some people may search for this, and some people may search for A329080.

			Sum_{k>=1} 1/(k^2-5) = -0.66683259566274485298...

Cf. A329080 (F(-5)), A329081 (F(-3)), A329082 (F(-2)), A113319 (F(1)), A329083 (F(2)), A329084 (F(3)), A329085 (F(4)), A329086 (F(5)).

Cf. this sequence (f(-5)), A329088 (f(-3)), A329089 (f(-2)), A013661 (f(0)), A259171 (f(1)), A329090 (f(2)), A329091 (f(3)), A329092 (f(4)), A329093 (f(5)).

RealDigits[(1 - Sqrt[5]*Pi*Cot[Sqrt[5]*Pi])/10, 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

default(realprecision, 100); my(f(x) = (-1 + (sqrt(x)*Pi)/tanh(sqrt(x)*Pi))/(2*x)); f(-5)

sumnumrat(1/(x^2-5), 1) \\ Charles R Greathouse IV, Jan 20 2022

Sum_{k>=1} 1/(k^2-5) = (-1 + (sqrt(-5)*Pi)*coth(sqrt(-5)*Pi))/(-10) = (-1 + (sqrt(5)*Pi)*cot(sqrt(5)*Pi))/(-10).

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A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

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A071253 a(n) = n^2*(n^2+1).

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A329083 Decimal expansion of Sum_{k>=0} 1/(k^2+2).

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A329084 Decimal expansion of Sum_{k>=0} 1/(k^2+3).

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A329085 Decimal expansion of Sum_{k>=0} 1/(k^2+4).

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A329080 Decimal expansion of Sum_{k>=0} 1/(k^2-5), negated.

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A329081 Decimal expansion of Sum_{k>=0} 1/(k^2-3).