A282824 -id:A282824 - cp's OEIS frontend

A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

0, 16, 928, 119344, 3078464, 1132669904, 606887707616, 49610806397296, 48006150564413056, 48265162121607952, 8192066749392160288, 15200753287254377716912, 33677610844789597790454208

Offset: 1

Artur Jasinski, Mar 03 2010

All numbers in this sequence are divisible by 16. For A173953/16 see A173955.

a(n+2)/A173954(n+2) is, for n >= 0, the partial sum Sum_{k=0..n} 1/(k + 3/4)^2 = 16*Sum_{k=0..n} 1/(4*k + 3)^2. The limit n -> infinity is given in A282824 as Zeta(2, 3/4) = Psi(1, 3/4) = Pi^2 - 8*Catalan, with the trigamma function Psi(1, z) and the Catalan constant A006752.

			The rationals r(n) = Zeta(2, 3/4) - Zeta(2, n-1/4) begin:  0/1, 16/9, 928/441, 119344/53361, 3078464/1334025, 1132669904/481583025, 606887707616/254757420225, 49610806397296/20635351038225, ... - _Wolfdieter Lang_, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Trigamma Function

Denominators are in A173954.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173955.

[0] cat [Numerator((&+[16/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018

r := n -> Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4):
seq(numer(simplify(r(n))), n=1..13); # Peter Luschny, Nov 14 2017

Table[Numerator[FunctionExpand[Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4]]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[128*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[16*Sum[1/(4*k + 3)^2, {k, 0, n-1}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)

for(n=1,20, print1(numerator(16*sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

a(n) = Numerator of (Pi^2 - 8*Catalan - Zeta(2, (4 n - 1)/4)).
Numerator of 128*n*Sum_{k>=1} (4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2). - Vaclav Kotesovec, Nov 14 2017
Numerator of 16*Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. See a comment above. - Wolfdieter Lang, Nov 14 2017

Name simplified by Peter Luschny, Nov 14 2017

A282823 Decimal expansion of Pi^2 + 8*K, where K is Catalan's constant.

Original entry on oeis.org

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

Offset: 2

Bruno Berselli, Mar 06 2017

			17.19732915450711073927131911933522402150689440149416770054533433319414...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Polygamma Function (formula 24).
Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556.

Cf. A000796, A006752, A222183, A282824.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)^2 + 8*Catalan(R); // G. C. Greubel, Aug 24 2018

Psi(1,1/4) ; evalf(%) # R. J. Mathar, Dec 04 2024

RealDigits[Pi^2 + 8 Catalan, 10, 100][[1]]

Pi^2 + 8*Catalan \\ Charles R Greathouse IV, Mar 04 2018

zetahurwitz(2,1/4) \\ Charles R Greathouse IV, Mar 04 2018

Equals 16*A222183.
Equals Psi(1, 1/4), where Psi(r, x) is the Polygamma function of order r.
Equals Sum_{k>=0} 1/(k + 1/4)^2. - Amiram Eldar, May 17 2022

A375392 Decimal expansion of the hyperbolic volume of the link complement of the Borromean rings.

Original entry on oeis.org

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

Offset: 1

Luc Ta, Aug 21 2024

			7.32772475341775212043682811945907288619319499425337707...

Luc Ta, Table of n, a(n) for n = 1..10000
William P. Thurston, The Geometry and Topology of Three-Manifolds: With a Preface by Steven P. Kerckhoff, American Mathematical Society (2022): p. 165, Example.
Index entries for sequences related to knots

Cf. A006752, A247685, A091518, A282824.

Mathematica
```
RealDigits[N[8 * Catalan, 100]][[1]]
```

Equals 8*A006752 = 2*A247685.
Equals -16*Integral_{x=0..Pi/4} log|2*sin(x)| dx.
Equals 2*hyperbolic volume of the link complement of the Whitehead link.

A375772 Decimal expansion of the absolute value of the second derivative of the digamma function at 3/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 27 2024

			psi''(3/4) = -5.3026332163376396314327...

Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95.

Cf. A200134 (psi(3/4)), A282824 (psi'(3/4)).

Maple
```
2*(Pi^3-28*Zeta(3)); evalf(%) ;
```

RealDigits[PolyGamma[2, 3/4], 10, 105][[1]] (* Vaclav Kotesovec, Aug 27 2024 *)

psi''(3/4) = 2*(Pi^3-28*zeta(3)).

cp's OEIS Frontend

A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.

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A282823 Decimal expansion of Pi^2 + 8*K, where K is Catalan's constant.

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A375392 Decimal expansion of the hyperbolic volume of the link complement of the Borromean rings.

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A375772 Decimal expansion of the absolute value of the second derivative of the digamma function at 3/4.

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