A233091 -id:A233091 - cp's OEIS frontend

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

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

Offset: 0

N. J. A. Sloane

Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015
a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]
Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - Reinhard Zumkeller, Sep 29 2008
A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009
Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012
a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013
Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]
The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020
For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Eric Weisstein's World of Mathematics, Kronecker Symbol
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
Index entries for "core" sequences
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,1).

Ones complement of A059841.
Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
Cf. A154955 (Mobius transform), A131577 (binomial transform).
Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
Parity of A005811.

a000035 n = n `mod` 2  -- James Spahlinger, Oct 08 2012

a000035_list = cycle [0,1]  -- Reinhard Zumkeller, Jan 06 2012

A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];

PadLeft[{},110,{0,1}] (* Harvey P. Dale, Sep 25 2011 *)

A000035(n):=mod(n,2)$
makelist(A000035(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=n%2;
```

a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* Ralf Stephan, Mar 27 2015 */

def A000035(n): return n & 1 # Chai Wah Wu, May 25 2022

(define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Mar 21 2017

a(n) = (1 - (-1)^n)/2.
a(n) = n mod 2.
a(n) = 1 - a(n-1).
Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001
G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003
a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005
Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012
a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013
a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015
Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015
a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015
Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024

A002117 Apéry's number or Apéry's constant zeta(3). Decimal expansion of zeta(3) = Sum_{m >= 1} 1/m^3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Sometimes called Apéry's constant.
"A natural question is whether Zeta(3) is a rational multiple of Pi^3. This is not known, though in 1978 R. Apéry succeeded in proving that Zeta(3) is irrational. In Chapter 8 we pointed out that the probability that two random integers are relatively prime is 6/Pi^2, which is 1/Zeta(2). This generalizes to: The probability that k random integers are relatively prime is 1/Zeta(k) ... ." [Stan Wagon]
In 2001 Tanguy Rivoal showed that there are infinitely many odd (positive) integers at which zeta is irrational, including at least one value j in the range 5 <= j <= 21 (refined the same year by Zudilin to 5 <= j <= 11), at which zeta(j) is irrational. See the Rivoal link for further information and references.
The reciprocal of this constant is the probability that three integers chosen randomly using uniform distribution are relatively prime. - Joseph Biberstine (jrbibers(AT)indiana.edu), Apr 13 2005
Also the value of zeta(1,2), the double zeta-function of arguments 1 and 2. - R. J. Mathar, Oct 10 2011
Also the length of minimal spanning tree for large complete graph with uniform random edge lengths between 0 and 1, cf. link to John Baez's comment. - M. F. Hasler, Sep 26 2017
Sum of the inverses of the cubes (A000578). - Michael B. Porter, Nov 27 2017
This number is the average value of sigma_2(n)/n^2 where sigma_2(n) is the sum of the squares of the divisors of n. - Dimitri Papadopoulos, Jan 07 2022

			1.2020569031595942853997...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 261.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 40-53, 500.
A. Fletcher, J. C. P. Miller, L. Rosenhead, and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 84.
R. William Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, Computers in Mathematics (Stanford CA, 1986); Lecture Notes in Pure and Appl. Math., Dekker, New York, 125 (1990), 261-284; MR 91h:11154.
Xavier Gourdon, Analyse, Les Maths en tête, Ellipses, 1994, Exemple 3, page 224.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F17, Series associated with the zeta-function, p. 391.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press; 6 edition (2008), pp. 47, 268-269.
Paul Levrie, The Ubiquitous Apéry Number, Math. Intelligencer, Vol. 45, No. 2, 2023, pp. 118-119.
A. A. Markoff, Mémoire sur la transformation de séries peu convergentes en séries très convergentes, Mém. de l'Acad. Imp. Sci. de St. Pétersbourg, XXXVII, 1890.
Paul J. Nahin, In Pursuit of Zeta-3, Princeton University Press, 2021.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stan Wagon, Mathematica In Action, W. H. Freeman and Company, NY, 1991, page 354.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 33.
A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Dover (1987), Ex. 92-93.

Harry J. Smith, Table of n, a(n) for n = 1..20002
T. Amdeberhan, Faster and Faster convergent series for zeta(3), arXiv:math/9804126 [math.CO], 1998.
Kunihiro Aoki and Ryo Furue, A model for the size distribution of marine microplastics: a statistical mechanics approach, arXiv:2103.10221 [physics.ao-ph], 2021.
Peter Bala, New series for old functions.
Peter Bala, Some series for zeta(3), Nov 2023.
John Baez, Comments about zeta(3), Azimuth Project blog, August 2017.
R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (7), (9), (19).
F. Beukers, A Note on the Irrationality of zeta(2) and zeta(3), Bull. London Math. Soc., 11 (3) (1979): 268-272.
J. Borwein and D. Bradley, Empirically determined Apéry-like formulas for zeta(4n+3), arXiv:math/0505124 [math.CA], 2005.
Mainendra Kumar Dewangan and Subhra Datta, Effective permeability tensor of confined flows with wall grooves of arbitrary shape, J. of Fluid Mechanics (2020) Vol. 891.
Dr. Math, Probability of Random Numbers Being Coprime.
L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
L. Euler, De summis serierum reciprocarum, E41.
X. Gourdon and P. Sebah, The Apery's constant: zeta(3).
Brady Haran and Tony Padilla, Apéry's constant (calculated with Twitter), Numberphile video (2017).
W. Janous, Around Apéry's constant, J. Inequ. Pure Appl. Math. 7(1) (2006), #35.
Yasuyuki Kachi and Pavlos Tzermias, Infinite products involving zeta(3) and Catalan's constant, Journal of Integer Sequences, 15 (2012), #12.9.4.
Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.
M. Kondratiewa and S. Sadov, Markov's transformation of series and the WZ method, arXiv:math/0405592 [math.CA], 2004.
Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
John Landen, Mathematical memoirs respecting a variety of subjects Vol. I, London, 1780.
F. M. S. Lima, Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study, arXiv:0910.2684 [math.NT], 2009-2012.
Jonah Lissner, Theoretical Physics Utilizations Of Riemann Zeta Function Odd Positive Integer Three, ResearchGate (2024).
Junqi Liu, Jujian Zhang, and Lihong Zhi, A formal proof of the irrationality of ζ(3) in Lean 4, arXiv preprint (2025). arXiv:2503.07625 [math.NT]
C. Lupu and D. Orr, Series representations for the Apéry constant zeta(3) involving the values zeta(2n), Ramanujan J. 48(3) (2019), 477-494.
R. J. Mathar, Yet another table of integrals, arXiv:1207.5845 [math.CA], 2012-2014.
G. P. Michon, Roger Apéry, Numericana.
S. D. Miller, An Easier Way to Show zeta(3) is Irrational.
Michael Penn, Euler's harmonic number identity, YouTube video, 2020.
Simon Plouffe, Zeta(3) or Apéry's constant to 2000 places.
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each (gzipped file).
A. van der Poorten, A Proof that Euler Missed.
Tanguy Rivoal, Irrationality of the zeta Function on Odd Integers [ps file].
Tanguy Rivoal, Irrationality of the zeta Function on Odd Integers [pdf file].
Ernst E. Scheufens, From Fourier series to rapidly convergent series for zeta(3), Mathematics Magazine, Vol. 84, No. 1 (2011), pp. 26-32.
G. Villemin's Almanach of Numbers, Constante d'Apéry (in French).
S. Wedeniwski, The value of zeta(3) to 1000000 places [Gutenberg Project Etext].
S. Wedeniwski, Plouffe's Inverter, Apery's constant to 128000026 decimal digits.
S. Wedeniwski, The value of zeta(3) to 1000000 decimal digits.
Eric Weisstein's World of Mathematics, Apéry's Constant.
Eric Weisstein's World of Mathematics, Relatively Prime.
Wikipedia, Riemann zeta function.
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.
J. W. Wrench, Jr., Letter to N. J. A. Sloane, Feb 04 1971.
Wenzhe Yang, Apéry's irrationality proof, mirror symmetry and Beukers' modular forms, arXiv:1911.02608 [math.NT], 2019.
Wadim Zudilin, An elementary proof of Apéry's theorem, arXiv:math/0202159 [math.NT], 2002.
Index entries for zeta function.

Cf. A013631, A013679, A013661, A013663, A013667, A013669, A013671, A013675, A013677, A059956 (6/Pi^2), A084225; A084226.
Cf. A197070: 3*zeta(3)/4; A233090: 5*zeta(3)/8; A233091: 7*zeta(3)/8.
Cf. A001008, A002805, A143003, A143007.
Cf. A000578 (cubes).
Cf. sums of inverses: A152623 (tetrahedral numbers), A175577 (octahedral numbers), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).

L:=RiemannZeta(: Precision:=100); Evaluate(L,3); // G. C. Greubel, Aug 21 2018

# Calculates an approximation with n exact decimal places (small deviation
# in the last digits are possible). Goes back to ideas of A. A. Markoff 1890.
zeta3 := proc(n) local s, w, v, k; s := 0; w := -1; v := 4;
for k from 2 by 2 to 7*n/2 do
    w := -w*v/k;
    v := v + 8;
    s := s + 1/(w*k^3);
od; 20*s; evalf(%, n) end:
zeta3(10000); # Peter Luschny, Jun 10 2020

RealDigits[ N[ Zeta[3], 100] ] [ [1] ]
(* Second program (historical interest): *)
d[n_] := 34*n^3 + 51*n^2 + 27*n + 5; 6/Fold[Function[d[#2-1] - #2^6/#1], 5, Reverse[Range[100]]] // N[#, 108]& // RealDigits // First
(* Jean-François Alcover, Sep 19 2014, after Apéry's continued fraction *)

fpprec : 100$ ev(bfloat(zeta(3)))$ bfloat(%); /* Martin Ettl, Oct 21 2012 */

default(realprecision, 20080); x=zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002117.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009

from mpmath import mp, apery
mp.dps=109
print([int(z) for z in list(str(apery).replace('.', ''))[:-1]]) # Indranil Ghosh, Jul 08 2017

Lima gives an approximation to zeta(3) as (236*log(2)^3)/197 - 283/394*Pi*log(2)^2 + 11/394*Pi^2*log(2) + 209/394*log(sqrt(2) + 1)^3 - 5/197 + (93*Catalan*Pi)/197. - Jonathan Vos Post, Oct 14 2009 [Corrected by Wouter Meeussen, Apr 04 2010]
zeta(3) = 5/2*Integral_(x=0..2*log((1+sqrt(5))/2), x^2/(exp(x)-1)) + 10/3*(log((1+sqrt(5))/2))^3. - Seiichi Kirikami, Aug 12 2011
zeta(3) = -4/3*Integral_{x=0..1} log(x)/x*log(1+x) = Integral_{x=0..1} log(x)/x*log(1-x) = -4/7*Integral_{x=0..1} log(x)/x*log((1+x)/(1-x)) = 4*Integral_{x=0..1} 1/x*log(1+x)^2 = 1/2*Integral_{x=0..1} 1/x*log(1-x)^2 = -16/7*Integral_{x=0..Pi/2} x*log(2*cos(x)) = -4/Pi*Integral_{x=0..Pi/2} x^2*log(2*cos(x)). - Jean-François Alcover, Apr 02 2013, after R. J. Mathar
From Peter Bala, Dec 04 2013: (Start)
zeta(3) = (16/7)*Sum_{k even} (k^3 + k^5)/(k^2 - 1)^4.
zeta(3) - 1 = Sum_{k >= 1} 1/(k^3 + 4*k^7) = 1/(5 - 1^6/(21 - 2^6/(55 - 3^6/(119 - ... - (n - 1)^6/((2*n - 1)*(n^2 - n + 5) - ...))))) (continued fraction).
More generally, there is a sequence of polynomials P(n,x) (of degree 2*n) such that
zeta(3) - Sum_{k = 1..n} 1/k^3 = Sum_{k >= 1} 1/( k^3*P(n,k-1)*P(n,k) ) = 1/((2*n^2 + 2*n + 1) - 1^6/(3*(2*n^2 + 2*n + 3) - 2^6/(5*(2*n^2 + 2*n + 7) - 3^6/(7*(2*n^2 + 2*n + 13) - ...)))) (continued fraction). See A143003 and A143007 for details.
Series acceleration formulas:
zeta(3) = (5/2)*Sum_{n >= 1} (-1)^(n+1)/( n^3*binomial(2*n,n) )
= (5/2)*Sum_{n >= 1} P(n)/( (2*n(2*n - 1))^3*binomial(4*n,2*n) )
= (5/2)*Sum_{n >= 1} (-1)^(n+1)*Q(n)/( (3*n(3*n - 1)*(3*n - 2))^3*binomial(6*n,3*n) ), where P(n) = 24*n^3 + 4*n^2 - 6*n + 1 and Q(n) = 9477*n^6 - 11421*n^5 + 5265*n^4 - 1701*n^3 + 558*n^2 - 108*n + 8 (Bala, section 7). (End)
zeta(3) = Sum_{n >= 1} (A010052(n)/n^(3/2)) = Sum_{n >= 1} ( (floor(sqrt(n)) - floor(sqrt(n-1)))/n^(3/2) ). - Mikael Aaltonen, Feb 22 2015
zeta(3) = Product_{k>=1} 1/(1 - 1/prime(k)^3). - Vaclav Kotesovec, Apr 30 2020
zeta(3) = 4*(2*log(2) - 1 - 2*Sum_{k>=2} zeta(2*k+1)/2^(2*k+1)). - Jorge Coveiro, Jun 21 2020
zeta(3) = (4*zeta'''(1/2)*(zeta(1/2))^2-12*zeta(1/2)*zeta'(1/2)*zeta''(1/2)+8*(zeta'(1/2))^3-Pi^3*(zeta(1/2))^3)/(28*(zeta(1/2))^3). - Artur Jasinski, Jun 27 2020
zeta(3) = Sum_{k>=1} H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 31 2020
From Artur Jasinski, Sep 30 2020: (Start)
zeta(3) = (5/4)*Li_3(1/f^2) + Pi^2*log(f)/6 - 5*log(f)^3/6,
zeta(3) = (8/7)*Li_3(1/2) + (2/21)*Pi^2 log(2) - (4/21) log(2)^3, where f is golden ratio (A001622) and Li_3 is the polylogarithm function, formulas published by John Landen in 1780, p. 118. (End)
zeta(3) = (1/2)*Integral_{x=0..oo} x^2/(e^x-1) dx (Gourdon). - Bernard Schott, Apr 28 2021
From Peter Bala, Jan 18 2022: (Start)
zeta(3) = 1 + Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)) = 25/24 + (2!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*(4*n^4 + 2^4)) = 28333/27000 + (3!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*(4*n^4 + 2^4)*(4*n^4 + 3^4)). In general, for k >= 1, we have zeta(3) = r(k) + (k!)^4*Sum_{n >= 1} 1/(n^3*(4*n^4 + 1)*...*(4*n^4 + k^4)), where r(k) is rational.
zeta(3) = (6/7) + (64/7)*Sum_{n >= 1} n/(4*n^2 - 1)^3.
More generally, for k >= 0, it appears that zeta(3) = a(k) + b(k)*Sum_{n >= 1} n/( (4*n^2 - 1)*(4*n^2 - 9)*...*(4*n^2 - (2*k+1)^2) )^3, where a(k) and b(k) are rational.
zeta(3) = (10/7) - (128/7)*Sum_{n >= 1} n/(4*n^2 - 1)^4.
More generally, for k >= 0, it appears that zeta(3) = c(k) + d(k)*Sum_{n >= 1} n/( (4*n^2 - 1)*(4*n^2 - 9)*...*(4*n^2 - (2*k+1)^2) )^4, where c(k) and d(k) are rational. [added Nov 27 2023: for the values of a(k), b(k), c(k) and d(k) see the Bala 2023 link, Sections 8 and 9.]
zeta(3) = 2/3 + (2^13)/(3*7)*Sum_{n >= 1} n^3/(4*n^2 - 1)^6. (End)
zeta(3) = -Psi(2)(1/2)/14 (the second derivative of digamma function evaluated at 1/2). - Artur Jasinski, Mar 18 2022
zeta(3) = -(8*Pi^2/9) * Sum_{k>=0} zeta(2*k)/((2*k+1)*(2*k+3)*4^k) = (2*Pi^2/9) * (log(2) + 2 * Sum_{k>=0} zeta(2*k)/((2*k+3)*4^k)) (Scheufens, 2011, Glasser Math. Comp. 22 1968). - Amiram Eldar, May 28 2022
zeta(3) = Sum_{k>=1} (30*k-11) / (4*(2k-1)*k^3*(binomial(2k,k))^2) (Gosper, 1986 and Richard K. Guy reference). - Bernard Schott, Jul 20 2022
zeta(3) = (4/3)*Integral_{x >= 1} x*log(x)*(1 + log(x))*log(1 + 1/x^x) dx = (2/3)*Integral_{x >= 1} x^2*log(x)^2*(1 + log(x))/(1 + x^x) dx. - Peter Bala, Nov 27 2023
zeta_3(n) = 1/180*(-360*n^3*f(-3, n/4) + Pi^3*(n^4 + 20*n^2 + 16))/(n*(n^2 + 4)), where f(-3, n) = Sum_{k>=1} 1/(k^3*(exp(Pi*k/n) - 1)). Will give at least 1 digit of precision/term, example: zeta_3(5) = 1.202056944732.... - Simon Plouffe, Dec 21 2023
zeta(3) = 1 + (1/2)*Sum_{n >= 1} (2*n + 1)/(n^3*(n + 1)^3) = 5/4 - (1/4)*Sum_{n >= 1} (2*n + 1)/(n^4*(n + 1)^4) = 147/120 + (2/15)*Sum_{n >= 1} (2*n + 1)/(n^5*(n + 1)^5) - (64/15)*Sum_{n >= 1} (n + 1)/(n^5*(n + 2)^5) = 19/16 + (128/21)*Sum_{n >= 1} (n + 1)/(n^6*(n + 2)^6) - (1/21)*Sum_{n >= 1} (2*n + 1)/(n^6*(n + 1)^6). - Peter Bala, Apr 15 2024
Equals 7*Pi^3/180 - 2*Sum_{k>=1} 1/(k^3*(exp(2*Pi*k) - 1)) [Grosswald] (see Finch). - Stefano Spezia, Nov 01 2024
Equals 10*Integral_{x=0..1/2} arcsinh(x)^2/x dx = -5*Integral_{x=0..2*log(phi)} x*log(2*sinh(x/2))dx [Munthe Hjortnaes] (see Finch). - Stefano Spezia, Nov 03 2024
Equals Li_3(1) = Integral_{x=0..1} Li_2(x)/x dx = Integral_{x=0..1} Integral_{y=0..1} Li_1(xy)/xy dydx = Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} Li_0(xyz)/xyz dzdydx (see Beukers), in general Integral_{x_1,...,x_k=0..1} Li_{3-k}(Product_{n=1..k} x_n)/(Product_{n=1..k} x_n) dx_k...dx_1 = zeta(3), for any k > 0. - Miko Labalan, Dec 23 2024
zeta(3) = (1/2)*Sum_{m >= 1}(Sum_{n >= 1} 1/(m*n*(m+n))). - Ricardo Bittencourt, Feb 24 2025
zeta(3) = Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} 1/(1 - x*y*z) dz dy dx. - Kritsada Moomuang, May 22 2025
zeta(3) = Sum_{i, j >= 1} 1/(i^2*j*binomial(i+j, i)) = Sum_{k >= 1} 1/(k + 1)^2 * Sum_{j = 1..k} 1/j = zeta(2, 1) (multiple zeta value due to Euler). - Peter Bala, Aug 05 2025

More terms from David W. Wilson
Additional comments from Robert G. Wilson v, Dec 08 2000
Quotation from Stan Wagon corrected by N. J. A. Sloane on Dec 24 2005. Thanks to Jose Brox for noticing this error.
Edited by M. F. Hasler, Sep 26 2017

A153071 Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.

Original entry on oeis.org

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

Offset: 0

Stuart Clary, Dec 17 2008

			L(3, chi4) = Pi^3/32 = 0.9689461462593693804836348458469186...

Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, 1989. See page 293, Entry 25 (iii).
Leonhard Euler, Introductio in Analysin Infinitorum, First Part, Articles 175, 284 and 287.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.

J. T. Groenman, Problem 1511, Crux Mathematicorum, Vol. 16, No. 2 (1990), p. 43; Solution to Problem 1511, by Beatriz Margolis, ibid., Vol. 17, No. 3 (1991), pp. 92-93.
Qing-Hu Hou and Zhi-Wei Sun, A q-analogue of the identity Sum_{k>=0}(-1)^k/(2k+1)^3 = Pi^3/32, arXiv:1808.04717 [math.CO], 2018.
Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.
Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, section 2.2 entry L(m=4,r=2,s=3).
Michael Penn, An infinite tangent product, YouTube video, 2020.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.
Index entries for transcendental numbers.

Cf. A101455, A153072, A153073, A153074.
Cf. A233091, A251809, A331095.
Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

nmax = 1000; First[ RealDigits[Pi^3/32, 10, nmax] ]

PARI
```
Pi^3/32 \\ Michel Marcus, Aug 15 2018
```

chi4(k) = Kronecker(-4, k); chi4(k) is 0, 1, 0, -1 when k reduced modulo 4 is 0, 1, 2, 3, respectively; chi4 is A101455.
Series: L(3, chi4) = Sum_{k>=1} chi4(k) k^{-3} = 1 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 - 1/11^3 + 1/13^3 - 1/15^3 + ...
Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) Pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]
Closed form: L(3, chi4) = Pi^3/32 = 1/A331095.
Equals Sum_{n>=0} (-1)^n/(2*n+1)^3. - Jean-François Alcover, Mar 29 2013
Equals Product_{k>=3} (1 - tan(Pi/2^k)^4) (Groenman, 1990). - Amiram Eldar, Apr 03 2022
Equals Integral_{x=0..1} arcsinh(x)*arccos(x)/x dx (Kobayashi, 2021). - Amiram Eldar, Jun 23 2023
From Amiram Eldar, Nov 06 2023: (Start)
Equals beta(3), where beta is the Dirichlet beta function.
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^3)^(-1). (End)

Offset corrected by R. J. Mathar, Feb 05 2009

A197070 Decimal expansion of the Dirichlet eta-function at 3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 09 2011

This constant is irrational by Apéry's theorem. - Charles R Greathouse IV, Feb 11 2024

			0.9015426773696957140498036211335874930737...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table II (4)
Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), series last equation.
Seán Stewart, Problem 12206, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.
Wikipedia, Dirichlet eta function.

Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.

Maple
```
3*Zeta(3)/4 ; evalf(%) ;
```

RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* Bruno Berselli, Dec 20 2011 *)

-polylog(3,-1) \\ Charles R Greathouse IV, Mar 28 2012

3/4*zeta(3) \\ Charles R Greathouse IV, Mar 28 2012

Equals 3*zeta(3)/4 = 3*A002117/4.
Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - Jean-François Alcover, Nov 24 2014
Equals Sum_{n>=1} (-1)^(n+1)/n^3. - Terry D. Grant, Aug 03 2016
Equals Lim_{n -> infinity} A136675(n)/A334582(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - Amiram Eldar, Oct 04 2021
Equals -int_0^1 log(x)log(1+x)/x dx [Barbieri] - R. J. Mathar, Jun 07 2024

A087003 a(2n) = 0 and a(2n+1) = mu(2n+1); also the sum of Mobius function values computed for terms of 3x+1 trajectory started at n, provided that Collatz conjecture is true.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1

Offset: 1

Labos Elemer, Oct 02 2003

Observe that (these summatory) terms are from {-1,0,1}, so behave like Mobius function values, not like Mertens function values. Moreover, empirically: a(n) deviates from mu(initial-value) = mu(n) only if iv = n is an even squarefree number (i.e., it is from A039956). - This comment, like also the next one, concerns the original Collatz-related definition of this sequence. - Antti Karttunen, Sep 18 2017
From Marc LeBrun, Feb 19 2004: (Start)
Absolute values are the same as those of A091069. First consider the descending parts of Collatz (or 3x+1) trajectories, those that begin with even numbers 2^p k, with k odd. These go 2^p*k, 2^(p-1)*k, ... 2k, k. All but 2k and k are divisible by 4, a (rational) square, hence their mu values are all 0 and so they contribute nothing to the sum.
Then at the end, since mu(2k) = -mu(k), the last two steps cancel each other out. So every descending chain in a trajectory contributes 0. Of course the full trajectory of every even number consists entirely of descending chains, so A087003 is 0 for all even n.
On the other hand, the trajectory of every odd number consists of just that number followed by the trajectory of an even number (which contributes nothing) so A087003 is indeed equal to mu(n) for odd n.
(End)
The sequence is multiplicative; it may be defined as the Dirichlet inverse of the integers modulo 2 (A000035). - Gerard P. Michon, Apr 29 2007
a(n) appears in the second column of A156241 at every second row. - Mats Granvik, Feb 07 2009

Antti Karttunen, Table of n, a(n) for n = 1..10000
G. P. Michon, The Collatz problem.
G. P. Michon, Multiplicative functions.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A008683, A039956, A292273, A099990, A209229.

Cf. A000035 (the Dirichlet inverse), A318657/A318658 (the "Dirichlet Square Root").

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] lf[x_] := Length[fpl[x]] Table[Apply[Plus, Table[MoebiusMu[Part[fpl[w], j]], {j, 1, lf[w]}]], {w, 1, 256}]
Riffle[MoebiusMu[Range[1,121,2]],0] (* Harvey P. Dale, Jan 24 2025 *)

A006370(n) = if(n%2, 3*n+1, n/2); \\ This function from Michael B. Porter, May 29 2010
A087003(n) = { my(s=1); while(n>1, s += moebius(n); n = A006370(n)); (s); }; \\ Antti Karttunen, Sep 14 2017

a(n)={sumdiv(n, d,  my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 04 2018

A087003(n) = ((n%2)*moebius(n)); \\ Antti Karttunen, Sep 01 2018

a(n) = A008683(n) + A292273(n). - Antti Karttunen, Sep 14 2017
Moebius transform of A209229. - Andrew Howroyd, Aug 04 2018
From Jianing Song, Aug 04 2018: (Start)
Multiplicative with a(2^e) = 0, a(p^e) = (-1 + (-1)^e)/2 for odd primes p.
Dirichlet g.f.: 1/((1 - 2^(-s))*zeta(s)).
(End)
From Antti Karttunen, Sep 01 2018: (Start)
a(n) = A000035(n)*A008683(n).
Dirichlet convolution of A318657/A046644 with itself.
(End)
Sum_{n>=1} a(n)/n^2 = A217739 . Sum_{n>=1} a(n)/n^3 = A233091. Sum_{n>=1} a(n)/n^4 = A300707. - R. J. Mathar, Dec 17 2024

a(2n) = 0, a(2n+1) = mu(2n+1) added to the name as the new primary definition by Antti Karttunen, Sep 18 2017

A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033

			0.7512855644747464283748363509446562442281164327128118011201697220886...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)
Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), page 32.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Equals 5*zeta(3)/8.
Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023
Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024

A251809 Decimal expansion of 3sqrt(2)Pi^3/128.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 10 2014

Equals the value of the Dirichlet L-series of the non-principal character modulo 8 (A188510) at s=3. - Jianing Song, Nov 16 2019

			1.027722585936858567879256618002255767210100318536997465331084755185...

L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 64 (formula 340).

G. C. Greubel, Table of n, a(n) for n = 1..10000
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=3).
Index entries for transcendental numbers.

Cf. A153071: Sum_{i >= 0} (-1)^i/(2i+1)^3.
Cf. A233091: Sum_{i >= 0} 1/(2i+1)^3.
Cf. A093954, A188510.

R:= RealField(); 3*Sqrt(2)*Pi(R)^3/128; // G. C. Greubel, Jul 27 2018

RealDigits[3 Sqrt[2] Pi^3/128, 10, 90][[1]]

3*sqrt(2)*Pi^3/128 \\ G. C. Greubel, Jul 27 2018

Equals Sum_{i >= 0} (-1)^floor(i/2)/(2i+1)^3 = +1 +1/3^3 -1/5^3 -1/7^3 +1/9^3 +1/11^3 - ...
Equals Sum_{i >= 1} A188510(i)/i^3 = Sum_{i >= 1} Kronecker(-8,i)/i^3. - Jianing Song, Nov 16 2019
Equals 1/(Product_{p prime == 1 or 3 (mod 8)} (1 - 1/p^3) * Product_{p prime == 5 or 7 (mod 8)} (1 + 1/p^3)). - Amiram Eldar, Dec 17 2023

A352048 Sum of the squares of the divisor complements of the odd proper divisors of n.

Original entry on oeis.org

0, 4, 9, 16, 25, 40, 49, 64, 90, 104, 121, 160, 169, 200, 259, 256, 289, 364, 361, 416, 499, 488, 529, 640, 650, 680, 819, 800, 841, 1040, 961, 1024, 1219, 1160, 1299, 1456, 1369, 1448, 1699, 1664, 1681, 2000, 1849, 1952, 2365, 2120, 2209, 2560, 2450, 2604, 2899, 2720

Offset: 1

Wesley Ivan Hurt, Mar 01 2022

			a(10) = 10^2 * Sum_{d|10, d<10, d odd} 1 / d^2 = 10^2 * (1/1^2 + 1/5^2) = 104.

Robert Israel, Table of n, a(n) for n = 1..10000

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: A091954 (k=0), A352047 (k=1), this sequence (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Cf. A006519, A050999, A076577, A233091.

f:= proc(n) local m,d;
      m:= n/2^padic:-ordp(n,2);
      add((n/d)^2, d = select(`<`,numtheory:-divisors(m),n))
end proc:
map(f, [$1..60]); # Robert Israel, Apr 03 2023

a[n_] := n^2 DivisorSum[n, If[# < n && OddQ[#], 1/#^2, 0]&];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)
a[n_] := DivisorSigma[-2, n/2^IntegerExponent[n, 2]] * n^2 - Mod[n, 2]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = n^2*sumdiv(n, d, if ((dMichel Marcus, May 11 2023

a(n) = n^2 * sigma(n >> valuation(n, 2), -2) - n % 2; \\ Amiram Eldar, Oct 13 2023

a(n) = n^2 * Sum_{d|n, d

G.f.: Sum_{k>=2} k^2 * x^k / (1 - x^(2*k)). - Ilya Gutkovskiy, May 14 2023

From Amiram Eldar, Oct 13 2023: (Start)

a(n) = A050999(n) * A006519(n)^2 - A000035(n).

Sum_{k=1..n} a(k) = c * n^3 / 3, where c = 7*zeta(3)/8 = 1.0517997... (A233091). (End)

A276712 Decimal expansion of zeta(3)/8.

Original entry on oeis.org

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

Offset: 0

Terry D. Grant, Sep 15 2016

			0.150257112894949285674967270188...

James Dodson, The Mathematical Repository Containing Analytical Solutions of Five Hundred Questions: Mostly Selected from Scarce and Valuable Authors, (1748), page 375.

G. C. Greubel, Table of n, a(n) for n = 0..5000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864, table II (5)
Nick Lord, Problen 89.D, Problem Corner, The Mathematical Gazette, Vol. 89, No. 514 (2005), pp. 115-119; Solution, ibid., Vol. 89, No. 516 (2005), pp. 539-542.
Michael Penn, The solution is an important constant, YouTube video, 2021.

Cf. A001008, A002117, A002805, A013661, A111003, A222171, A233091.

SetDefaultRealField(RealField(120)); L:=RiemannZeta();  Evaluate(L,3)/8; // G. C. Greubel, Nov 24 2021

Mathematica
```
RealDigits[(Zeta[3])/8, 10, 100][[1]]
```

zeta(3)/8 \\ Michel Marcus, Sep 16 2016

Sage
```
(zeta(3)/8).n(100)
```

Equals Sum_{n>=1} 1/(2n)^3 = 1/8 + 1/64 + 1/216 + 1/512 + ...
Equals A002117/8.
zeta(3)/8 + A233091 = Sum_{n>=1} 1/(2n+1)^3 + Sum_{n>=1} 1/(2n)^3 = zeta(3).
Equals Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jul 22 2020
Equals Integral_{x=0..Pi/4} log(sin(x))*log(cos(x))/(sin(x)*cos(x)) dx (Lord, 2005). - Amiram Eldar, Jun 23 2023
Equals -integral_{x=0..1} log(x) log(1+x)/(1+x). [Barbieri] - R. J. Mathar, Jun 07 2024

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.0207800444333631028232547399039818253534109375...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 52.

Cf. A002117, A002194, A016777, A016779, A091925, A233091, A377558, A377559, A377560.

RealDigits[2Pi^3/(81Sqrt[3])+13Zeta[3]/27,10,100][[1]]

Equals Sum_{k>=0} 1/(3*k + 1)^3 (see Finch).
Equals -psi''(1/3)/54 (see Shamos).
Equals hypergeom([1/3, 1/3, 1/3, 1], [4/3, 4/3, 4/3], 1). - R. J. Mathar, Jul 14 2025

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cp's OEIS Frontend

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

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A002117 Apéry's number or Apéry's constant zeta(3). Decimal expansion of zeta(3) = Sum_{m >= 1} 1/m^3.

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A153071 Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.

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A197070 Decimal expansion of the Dirichlet eta-function at 3.

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A087003 a(2n) = 0 and a(2n+1) = mu(2n+1); also the sum of Mobius function values computed for terms of 3x+1 trajectory started at n, provided that Collatz conjecture is true.

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A251809 Decimal expansion of 3sqrt(2)Pi^3/128.

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.