A086731 -id:A086731 - cp's OEIS frontend

A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

This number is L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3, A102283. - Stuart Clary, Dec 17 2008
Equals 1/1^2 -1/2^2 +1/4^2 -1/5^2 +1/7^2 -1/8^2 +1/10^2 -1/11^2 +-... . This can be split as (1/1^2 -1/5^2 +1/7^2 -1/11^2 +-...) - (1/2^2 -1/4^2 +1/8^2 -1/10^2..) = (g(1)-g(5)) - (g(2)-g(4)). The first of these two series is A214552 and the second series is 1/(2^2)*(1-1/2^2 +1/4^2-1/5^2+-...), namely a quarter of the original series. Therefore 5/4 of this value here equals A214552. - R. J. Mathar, Jul 20 2012
Calegari, Dimitrov, & Tang prove that this number is irrational. - Charles R Greathouse IV, Aug 29 2024

			0.781302412896486296867...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Integrals of the Ising class, J. Phys. A 39 (2006) 12271, variable C_3.
Frank Calegari, Vesselin Dimitrov, and Yunqing Tang, The linear independence of 1, ζ(2), and L(2,χ₋₃), arXiv:2408.15403 [math.NT], 2024.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.
Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=3,r=2,s=2).
Lorenz Milla, World record computation of Dirichlet L(-3,2) series (1,000,000,000,000 digits)
Kh. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, El. J. Combin. 18 (2) (2012), Article P35, value of K after Theorem 4.

Cf. A086722-A086731, A102283, A214549 (principal character), A214552.

Cf. A153066, A153067, A153068.

nmax = 1000; First[ RealDigits[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, 10, nmax] ] (* Stuart Clary, Dec 17 2008 *)

zetahurwitz(2,1/3)/9 - zetahurwitz(2,2/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

From Jean-François Alcover, Jul 17 2014, updated Jan 23 2015: (Start)
Equals Sum_{n>=1} jacobi(-3, n+3)/n^2.
Equals (8/15)*4F3(1/2,1,1,2; 5/4,3/2,7/4; 3/4), where 4F3 is the generalized hypergeometric function.
Equals 4*Pi*log(3)/(3*sqrt(3)) - 4*Integral_{0..1} log(x+1)/(x^2-x+1) dx. (End)
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^2)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^2)^(-1). - Amiram Eldar, Nov 06 2023

A016970 a(n) = (6*n + 5)^2.

Original entry on oeis.org

25, 121, 289, 529, 841, 1225, 1681, 2209, 2809, 3481, 4225, 5041, 5929, 6889, 7921, 9025, 10201, 11449, 12769, 14161, 15625, 17161, 18769, 20449, 22201, 24025, 25921, 27889, 29929, 32041, 34225, 36481, 38809, 41209, 43681, 46225, 48841, 51529

Offset: 0

N. J. A. Sloane

The product of 4 successive terms of an arithmetic progression + square of the common difference is a square: a(n) = the square arising as the sum of first four terms of an arithmetic progression + n^2 where 1 is the first term and n is the common difference. a(1) = 25 = 1*2*3*4+1 a(2) = 121 = 1*3*5*7 +2^2 a(3) = 289 = 1*4*7*10 + 3^2, etc. - Amarnath Murthy, Mar 25 2004
If Y is a fixed 2-subset of a (6n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
Sequence found by reading the line from 25 in the direction 25, 121,... in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, Jul 28 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016969 (6*n+5), A086731, A174371.

List([0..40],n->(6*n+5)^2); # Muniru A Asiru, Dec 06 2018

[(6*n+5)^2: n in [0..50]]; // Vincenzo Librandi, May 07 2011

[(6*n+5)^2$n=0..40]; # Muniru A Asiru, Dec 06 2018

Array[(6 # + 5)^2 &, 38, 0] (* or *)
CoefficientList[Series[(-25 - 46 x - x^2)/(x - 1)^3, {x, 0, 37}], x] (* Michael De Vlieger, Dec 06 2018 *)
CoefficientList[Series[E^x (25 + 96 x + 36 x^2), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 07 2018 *)

a(n)=(6*n+5)^2 \\ Charles R Greathouse IV, Jul 28 2016

s=((25+46*x+x^2)/(1-x)^3).series(x, 20); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 07 2018

G.f.: (25 + 46*x + x^2) / (1-x)^3. - R. J. Mathar, Mar 10 2011
a(n) = 24 * A000326(n+1) + 1. - Jean-Bernard François, Oct 12 2014
a(n) = 6*A033579(n+1) + 1. - Miquel Cerda, Jul 28 2016
E.g.f.: exp(x)*(25 + 96*x + 36*x^2). - Stefano Spezia, Dec 07 2018
a(n) = A003215(3*n+2) + A002378(3*n+2). - Klaus Purath, Jun 09 2020
Sum_{n>=0} 1/a(n) = A086731. - Amiram Eldar, Nov 17 2020

A086722 Decimal expansion of g(1)+g(2)-g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 31 2003

By summing over g(1)-g(5) and g(2)-g(4) separately we obtain A214552 for the first difference and a quarter of A086724 for the second difference. - R. J. Mathar, Jul 20 2012

2/3 times this constant equals A086724 [Bailey, Borwein and Crandall, 2006] - R. J. Mathar, Jul 20 2012

			1.1719536193447294453... = A214552 + A086724/4 = 1/1^2 +1/2^2 -1/4^2 -1/5^2 +1/7^2 +1/8^2 -1/10^2 -1/11^2 ++--....

L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall, Integrals of the Ising class, Journal of Physics A: Mathematical and General, Vol. 39, No. 40 (2006), 12271; alternative link.

Cf. A086723-A086731, A214552.

g[k_] := PolyGamma[1, k/6]/36; RealDigits[g[1] + g[2] - g[4] - g[5], 10, 99] // First (* Jean-François Alcover, Feb 12 2013 *)

Equals -Integral_{x=0..1} log(x)/(x^2-x+1) dx. - Jean-François Alcover, Aug 29 2014

Equals Integral_{x>=0} x/(exp(x) + exp(-x) - 1) dx. - Amiram Eldar, May 22 2023

A086729 Decimal expansion of Pi^2/72.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

The original name was: Decimal expansion of Sum_{m=0..infinity} 1/(6*m+3)^2.

			0.1370778389040188697...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Index entries for transcendental numbers

Cf. A086722, A086723, A086724, A086725, A086726, A086727, A086728, A086730, A086731, A111003, A353908.

evalf(Pi^2/72) ;  # R. J. Mathar, Dec 18 2010

PARI
```
Pi^2/72 \\ Omar E. Pol, May 12 2022
```

Equals A111003/9. - R. J. Mathar, Dec 18 2010
From Amiram Eldar, Jul 19 2020: (Start)
Sum_{k>=0} (1/(12*k+3)^2 + 1/(12*k+9)^2).
Equals Integral_{x=1..oo} log(1 + 1/x^6)/x dx. (End)
Equals A353908/2. - Omar E. Pol, May 12 2022

New name after R. J. Mathar's Maple program. - Omar E. Pol, May 12 2022

A086730 Decimal expansion of sum(1/(6*m+4)^2,m=0..infinity).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

			0.085107650259964...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Cf. A086722-A086731.

RealDigits[ PolyGamma[1, 2/3]/36, 10, 98] // First // Prepend[#, 0]& (* Jean-François Alcover, Feb 12 2013 *)

psi'(2/3)/36 \\ Charles R Greathouse IV, Jul 19 2013

.085107650259964... = Psi'(2/3)/36 where Psi'(x) denotes the first derivative of the digamma function - R. Piyo (nagoya314(AT)yahoo.com), Dec 12 2004

8*Zeta(2)/9 = A214550 +4*(this constant) with Zeta(2) = A013661. - R. J. Mathar, Sep 15 2012

A086723 Decimal expansion of 1/(g(1)+g(2)-g(4)-g(5)), where g(k) = sum(1/(6*m+k)^2,m=0..infinity).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

			.85327608831408080...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Cf. A086722-A086731.

g[k_] := PolyGamma[1, k/6]/36; First[ RealDigits[1/(g[1] + g[2] - g[4] - g[5]), 10, 99]] (* Jean-François Alcover, Feb 12 2013 *)

A086726 Decimal expansion of sum(1/(6*m)^2,m=1..infinity).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

			.0456926129680062899...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Cf. A086722-A086731.

Join[{0},RealDigits[Pi^2/216,10,120][[1]]] (* Harvey P. Dale, Apr 30 2015 *)

Pi^2/216.

A086727 Decimal expansion of sum(1/(6*m+1)^2,m=0..infinity).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 31 2003

			1.0366253636763794...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Cf. A086722-A086731.

RealDigits[N[PolyGamma[1, 1/6]/36, 100]][[1]]

A086728 Decimal expansion of sum(1/(6*m+2)^2,m=0..infinity).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

			.2804332534840859467...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Cf. A086722-A086731.

RealDigits[ PolyGamma[1, 1/3]/36, 10, 99] // First (* Jean-François Alcover, Feb 12 2013 *)

Equals A214550 divided by 4. - R. J. Mathar, Sep 15 2012

A086725 Decimal expansion of 1/(g(1)-g(2)+g(4)-g(5)), where g(k) = Sum_{m>=0} 1/(6*m+k)^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 31 2003

			1.2799141324711212...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

Cf. A086722-A086731.

g[k_] := PolyGamma[1, k/6]/36; RealDigits[ 1/(g[1] - g[2] + g[4] - g[5]), 10, 99] // First (* Jean-François Alcover, Feb 13 2013 *)

cp's OEIS Frontend

A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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A016970 a(n) = (6*n + 5)^2.

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A086722 Decimal expansion of g(1)+g(2)-g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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A086729 Decimal expansion of Pi^2/72.

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A086730 Decimal expansion of sum(1/(6*m+4)^2,m=0..infinity).

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A086723 Decimal expansion of 1/(g(1)+g(2)-g(4)-g(5)), where g(k) = sum(1/(6*m+k)^2,m=0..infinity).

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A086726 Decimal expansion of sum(1/(6*m)^2,m=1..infinity).

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