A197070 -id:A197070 - cp's OEIS frontend

A000578 The cubes: a(n) = n^3.

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507

Offset: 0

N. J. A. Sloane

a(n) is the sum of the next n odd numbers; i.e., group the odd numbers so that the n-th group contains n elements like this: (1), (3, 5), (7, 9, 11), (13, 15, 17, 19), (21, 23, 25, 27, 29), ...; then each group sum = n^3 = a(n). Also the median of each group = n^2 = mean. As the sum of first n odd numbers is n^2 this gives another proof of the fact that the n-th partial sum = (n(n + 1)/2)^2. - Amarnath Murthy, Sep 14 2002
Total number of triangles resulting from criss-crossing cevians within a triangle so that two of its sides are each n-partitioned. - Lekraj Beedassy, Jun 02 2004. See Propp and Propp-Gubin for a proof.
Also structured triakis tetrahedral numbers (vertex structure 7) (cf. A100175 = alternate vertex); structured tetragonal prism numbers (vertex structure 7) (cf. A100177 = structured prisms); structured hexagonal diamond numbers (vertex structure 7) (cf. A100178 = alternate vertex; A000447 = structured diamonds); and structured trigonal anti-diamond numbers (vertex structure 7) (cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Schlaefli symbol for this polyhedron: {4, 3}.
Least multiple of n such that every partial sum is a square. - Amarnath Murthy, Sep 09 2005
Draw a regular hexagon. Construct points on each side of the hexagon such that these points divide each side into equally sized segments (i.e., a midpoint on each side or two points on each side placed to divide each side into three equally sized segments or so on), do the same construction for every side of the hexagon so that each side is equally divided in the same way. Connect all such points to each other with lines that are parallel to at least one side of the polygon. The result is a triangular tiling of the hexagon and the creation of a number of smaller regular hexagons. The equation gives the total number of regular hexagons found where n = the number of points drawn + 1. For example, if 1 point is drawn on each side then n = 1 + 1 = 2 and a(n) = 2^3 = 8 so there are 8 regular hexagons in total. If 2 points are drawn on each side then n = 2 + 1 = 3 and a(n) = 3^3 = 27 so there are 27 regular hexagons in total. - Noah Priluck (npriluck(AT)gmail.com), May 02 2007
The solutions of the Diophantine equation: (X/Y)^2 - X*Y = 0 are of the form: (n^3, n) with n >= 1. The solutions of the Diophantine equation: (m^2)*(X/Y)^2k - XY = 0 are of the form: (m*n^(2k + 1), m*n^(2k - 1)) with m >= 1, k >= 1 and n >= 1. The solutions of the Diophantine equation: (m^2)*(X/Y)^(2k + 1) - XY = 0 are of the form: (m*n^(k + 1), m*n^k) with m >= 1, k >= 1 and n >= 1. - Mohamed Bouhamida, Oct 04 2007
Except for the first two terms, the sequence corresponds to the Wiener indices of C_{2n} i.e., the cycle on 2n vertices (n > 1). - K.V.Iyer, Mar 16 2009
Totally multiplicative sequence with a(p) = p^3 for prime p. - Jaroslav Krizek, Nov 01 2009
Sums of rows of the triangle in A176271, n > 0. - Reinhard Zumkeller, Apr 13 2010
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
Numbers n for which order of torsion subgroup t of the elliptic curve y^2 = x^3 - n is t = 2. - Artur Jasinski, Jun 30 2010
The sequence with the lengths of the Pisano periods mod k is 1, 2, 3, 4, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 16, 17, 6, 19, 20, ... for k >= 1, apparently multiplicative and derived from A000027 by dividing every ninth term through 3. Cubic variant of A186646. - R. J. Mathar, Mar 10 2011
The number of atoms in a bcc (body-centered cubic) rhombic hexahedron with n atoms along one edge is n^3 (T. P. Martin, Shells of atoms, eq. (8)). - Brigitte Stepanov, Jul 02 2011
The inverse binomial transform yields the (finite) 0, 1, 6, 6 (third row in A019538 and A131689). - R. J. Mathar, Jan 16 2013
Twice the area of a triangle with vertices at (0, 0), (t(n - 1), t(n)), and (t(n), t(n - 1)), where t = A000217 are triangular numbers. - J. M. Bergot, Jun 25 2013
If n > 0 is not congruent to 5 (mod 6) then A010888(a(n)) divides a(n). - Ivan N. Ianakiev, Oct 16 2013
For n > 2, a(n) = twice the area of a triangle with vertices at points (binomial(n,3),binomial(n+2,3)), (binomial(n+1,3),binomial(n+1,3)), and (binomial(n+2,3),binomial(n,3)). - J. M. Bergot, Jun 14 2014
Determinants of the spiral knots S(4,k,(1,1,-1)). a(k) = det(S(4,k,(1,1,-1))). - Ryan Stees, Dec 14 2014
One of the oldest-known examples of this sequence is shown in the Senkereh tablet, BM 92698, which displays the first 32 terms in cuneiform. - Charles R Greathouse IV, Jan 21 2015
From Bui Quang Tuan, Mar 31 2015: (Start)
We construct a number triangle from the integers 1, 2, 3, ... 2*n-1 as follows. The first column contains all the integers 1, 2, 3, ... 2*n-1. Each succeeding column is the same as the previous column but without the first and last items. The last column contains only n. The sum of all the numbers in the triangle is n^3.
Here is the example for n = 4, where 1 + 2*2 + 3*3 + 4*4 + 3*5 + 2*6 + 7 = 64 = a(4):
  1
  2  2
  3  3  3
  4  4  4  4
  5  5  5
  6  6
  7
(End)
For n > 0, a(n) is the number of compositions of n+11 into n parts avoiding parts 2 and 3. - Milan Janjic, Jan 07 2016
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
Number of inequivalent face colorings of the cube using at most n colors such that each color appears at least twice. - David Nacin, Feb 22 2017
Consider A = {a,b,c} a set with three distinct members. The number of subsets of A is 8, including {a,b,c} and the empty set. The number of subsets from each of those 8 subsets is 27. If the number of such iterations is n, then the total number of subsets is a(n-1). - Gregory L. Simay, Jul 27 2018
By Fermat's Last Theorem, these are the integers of the form x^k with the least possible value of k such that x^k = y^k + z^k never has a solution in positive integers x, y, z for that k. - Felix Fröhlich, Jul 27 2018

			For k=3, b(3) = 2 b(2) - b(1) = 4-1 = 3, so det(S(4,3,(1,1,-1))) = 3*3^2 = 27.
For n=3, a(3) = 3 + (3*0^2 + 3*0 + 3*1^2 + 3*1 + 3*2^2 + 3*2) = 27. - _Patrick J. McNab_, Mar 28 2016

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 191.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 43, 64, 81.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 292.
T. Aaron Gulliver, "Sequences from cubes of integers", International Mathematical Journal, 4 (2003), no. 5, 439 - 445. See http://www.m-hikari.com/z2003.html for information about this journal. [I expanded the reference to make this easier to find. - N. J. A. Sloane, Feb 18 2019]
J. Propp and A. Propp-Gubin, "Counting Triangles in Triangles", Pi Mu Epsilon Journal (to appear).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 6-7.
D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
H. Bottomley, Illustration of initial terms
British National Museum, Tablet 92698
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
Ralph Greenberg, Math For Poets
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets [Cached version at the Wayback Machine]
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75. - fixed by _Felix Fröhlich_, Jun 16 2014
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (8).
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
James Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 25 September 2024.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.
Eric Weisstein's World of Mathematics, Cubic Number, and Hex Pyramidal Number
Ronald Yannone, Hilbert Matrix Analyses
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for "core" sequences
Index entries for sequences related to Benford's law

(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
For sums of cubes, cf. A000537 (partial sums), A003072, A003325, A024166, A024670, A101102 (fifth partial sums).
Cf. A001158 (inverse Möbius transform), A007412 (complement), A030078(n) (cubes of primes), A048766, A058645 (binomial transform), A065876, A101094, A101097.
Subsequence of A145784.
Cf. A260260 (comment). - Bruno Berselli, Jul 22 2015
Cf. A000292 (tetrahedral numbers), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Cf. A098737 (main diagonal).

a000578 = (^ 3)
a000578_list = 0 : 1 : 8 : zipWith (+)
   (map (+ 6) a000578_list)
   (map (* 3) $ tail $ zipWith (-) (tail a000578_list) a000578_list)
-- Reinhard Zumkeller, Sep 05 2015, May 24 2012, Oct 22 2011

[ n^3 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014

I:=[0,1,8,27]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jul 05 2014

A000578 := n->n^3;
seq(A000578(n), n=0..50);
isA000578 := proc(r)
    local p;
    if r = 0 or r =1 then
        true;
    else
        for p in ifactors(r)[2] do
            if op(2, p) mod 3 <> 0 then
                return false;
            end if;
        end do:
        true ;
    end if;
end proc: # R. J. Mathar, Oct 08 2013

Table[n^3, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
CoefficientList[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jul 05 2014 *)
Accumulate[Table[3n^2+3n+1,{n,0,20}]] (* or *) LinearRecurrence[{4,-6,4,-1},{1,8,27,64},20](* Harvey P. Dale, Aug 18 2018 *)

A000578(n):=n^3$
makelist(A000578(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

A000578(n)=n^3 \\ M. F. Hasler, Apr 12 2008

is(n)=ispower(n,3) \\ Charles R Greathouse IV, Feb 20 2012

A000578_list, m = [], [6, -6, 1, 0]
for _ in range(10**2):
    A000578_list.append(m[-1])
    for i in range(3):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

(define (A000578 n) (* n n n)) ;; Antti Karttunen, Oct 06 2017

a(n) = Sum_{i=0..n-1} A003215(i).
Multiplicative with a(p^e) = p^(3e). - David W. Wilson, Aug 01 2001
G.f.: x*(1+4*x+x^2)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
Dirichlet generating function: zeta(s-3). - Franklin T. Adams-Watters, Sep 11 2005, Amarnath Murthy, Sep 09 2005
E.g.f.: (1+3*x+x^2)*x*exp(x). - Franklin T. Adams-Watters, Sep 11 2005 - Amarnath Murthy, Sep 09 2005
a(n) = Sum_{i=1..n} (Sum_{j=i..n+i-1} A002024(j,i)). - Reinhard Zumkeller, Jun 24 2007
a(n) = lcm(n, (n - 1)^2) - (n - 1)^2. E.g.: lcm(1, (1 - 1)^2) - (1 - 1)^2 = 0, lcm(2, (2 - 1)^2) - (2 - 1)^2 = 1, lcm(3, (3 - 1)^2) - (3 - 1)^2 = 8, ... - Mats Granvik, Sep 24 2007
Starting (1, 8, 27, 64, 125, ...), = binomial transform of [1, 7, 12, 6, 0, 0, 0, ...]. - Gary W. Adamson, Nov 21 2007
a(n) = A007531(n) + A000567(n). - Reinhard Zumkeller, Sep 18 2009
a(n) = binomial(n+2,3) + 4*binomial(n+1,3) + binomial(n,3). [Worpitzky's identity for cubes. See. e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n + 6*binomial(n+1,3) = binomial(n,1)+6*binomial(n+1,3). - Ron Knott, Jun 10 2019
A010057(a(n)) = 1. - Reinhard Zumkeller, Oct 22 2011
a(n) = A000537(n) - A000537(n-1), difference between 2 squares of consecutive triangular numbers. - Pierre CAMI, Feb 20 2012
a(n) = A048395(n) - 2*A006002(n). - J. M. Bergot, Nov 25 2012
a(n) = 1 + 7*(n-1) + 6*(n-1)*(n-2) + (n-1)*(n-2)*(n-3). - Antonio Alberto Olivares, Apr 03 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6. - Ant King Apr 29 2013
a(n) = A000330(n) + Sum_{i=1..n-1} A014105(i), n >= 1. - Ivan N. Ianakiev, Sep 20 2013
a(k) = det(S(4,k,(1,1,-1))) = k*b(k)^2, where b(1)=1, b(2)=2, b(k) = 2*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - Ryan Stees, Dec 14 2014
For n >= 1, a(n) = A152618(n-1) + A033996(n-1). - Bui Quang Tuan, Apr 01 2015
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Jon Tavasanis, Feb 21 2016
a(n) = n + Sum_{j=0..n-1} Sum_{k=1..2} binomial(3,k)*j^(3-k). - Patrick J. McNab, Mar 28 2016
a(n) = A000292(n-1) * 6 + n. - Zhandos Mambetaliyev, Nov 24 2016
a(n) = n*binomial(n+1, 2) + 2*binomial(n+1, 3) + binomial(n,3). - Tony Foster III, Nov 14 2017
From Amiram Eldar, Jul 02 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(3) (A002117).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/4 (A197070). (End)
From Amiram Eldar, Jan 20 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi.
Product_{n>=2} (1 - 1/a(n)) = cosh(sqrt(3)*Pi/2)/(3*Pi). (End)
a(n) = Sum_{d|n} sigma_3(d)*mu(n/d) = Sum_{d|n} A001158(d)*A008683(n/d). Moebius transform of sigma_3(n). - Ridouane Oudra, Apr 15 2021

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

A033999 a(n) = (-1)^n.

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

(-1)^(n+1) = signed area of parallelogram with vertices (0,0), U=(F(n),F(n+1)), V=(F(n+1),F(n+2)), where F = A000045 (Fibonacci numbers). The area of every such parallelogram is 1. The signed area is -1 if and only if F(n+1)^2 > F(n)*F(n+2), or, equivalently, n is even, or, equivalently, the vector U is "above" V, indicating that U and V "cross" as n -> n+1. - Clark Kimberling, Sep 09 2013
Periodic with period length 2. - Ray Chandler, Apr 03 2017
From Bernard Schott, May 11 2022: (Start)
Cesàro mean theorem: When a(n) has a limit (finite or infinite) in the usual sense, then c(n) = (a(1)+...+a(n))/n has the same Cesàro limit, but the converse is false. This sequence is a counterexample in the case of a finite Cesàro limit (see A237420 for counterexample with an infinite Cesàro limit).
This sequence is not convergent in the usual sense because a(2n) = 1 while a(2n+1) = -1; the successive arithmetic means c(n) of the first n terms of the sequence are 1/1, 0/2, 1/3, 0/4, 1/5, 0/6, ... so c(2n) = 1/(2n+1) and c(2n+1) = 0, hence the Cesàro limit is 0 because c(n) -> 0 when n -> oo.
In fact, when sequence a(n) is "Period k: [a1, a2, ..., ak]", then the Cesàro limit c of this sequence is (a1+a2+...+ak)/k.
Note that the converse of the theorem is true iff a(n) is monotonic (End).

			G.f. = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 + ...

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
S. K. Ghosal and J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.
Tanya Khovanova, Recursive Sequences
Mathematics Stack Exchange, Convergence of series implies convergence of Cesàro mean, 2013.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
ProofWiki, Cesàro mean.
Michael Somos, Rational Function Multiplicative Coefficients
Eric Weisstein's World of Mathematics, Inverse Tangent
Eric Weisstein's World of Mathematics, Stirling Transform
Wikipedia, Ernesto Cesàro.
Wikipedia, Grandi's series
Wikipedia, +/-1-sequence
Wikipedia, Dirichlet eta function
Wikipédia, Lemme de Cesàro (in French).
Index entries for linear recurrences with constant coefficients, signature (-1).
Index to sequences related to inverse of cyclotomic polynomials

Cf. A056594, A059841, A063007, A122803.
About Cesàro mean theorem: A114112, A237420.
Cf. A072691 (abs. val. Dgf at s=2), A197070 (abs. val. Dgf at s=3), A267315 (abs. val. Dgf at s=4).

a033999 = (1 -) . (* 2) . (`mod` 2)
a033999_list = cycle [1,-1] -- Reinhard Zumkeller, May 06 2012, Jan 02 2012

[(-1)^n : n in [0..100]]; // Wesley Ivan Hurt, Nov 19 2014

A033999 := n->(-1)^n: seq(A033999(n), n=0..100);

Table[(-1)^n, {n, 0, 88}] (* Alonso del Arte, Nov 30 2009 *)
PadRight[{}, 89, {1, -1}] (* Arkadiusz Wesolowski, Sep 16 2012 *)

a(n)=1-2*(n%2) /* Jaume Oliver Lafont, Mar 20 2009 */

def A033999(n): return -1 if n % 2 else 1 # Chai Wah Wu, May 24 2022

G.f.: 1/(1+x).
E.g.f.: exp(-x).
Linear recurrence: a(0)=1, a(n)=-a(n-1) for n>0. - Jaume Oliver Lafont, Mar 20 2009
Sum_{k=0..n} a(k) = A059841(n). - Jaume Oliver Lafont, Nov 21 2009
Sum_{k>=0} a(k)/(k+1) = log(2). - Jaume Oliver Lafont, Mar 30 2010
Euler transform of length 2 sequence [ -1, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 2 sequence [ -1, 2]. - Michael Somos, Mar 21 2011
a(n) = -b(n) where b(n) = multiplicative with b(2^e) = -1 if e>0, b(p^e) = 1 if p>2. - Michael Somos, Mar 21 2011
a(n) = a(-n) = a(n + 2) = cos(n * Pi). a(n) = c_2(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n) = (1/2)*Product_{k=0..2*n-1} 2*cos((2*k+1)*Pi/(4*n)), n >= 1. See the product given in the Oct 21 2013 formula comment in A056594, and replace there n -> 2*n. - Wolfdieter Lang, Oct 23 2013
D.g.f.: (2^(1-s)-1)*zeta(s) = -eta(s) (the Dirichlet eta function). - Ralf Stephan, Mar 27 2015
From Ilya Gutkovskiy, Aug 17 2016: (Start)
a(n) = T_n(-1), where T_n(x) are the Chebyshev polynomials of the first kind.
Binomial transform of A122803. (End)
a(n) = exp(i*Pi*n) = exp(-i*Pi*n). - Carauleanu Marc, Sep 15 2016
a(n) = Sum_{k=0..n} (-1)^k*A063007(n, k), n >= 0. - Wolfdieter Lang, Sep 13 2016

cp's OEIS Frontend

A000578 The cubes: a(n) = n^3.

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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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A033999 a(n) = (-1)^n.

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A233091 Decimal expansion of Sum_{i>=0} 1/(2*i+1)^3.

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A267315 Decimal expansion of the Dirichlet eta function at 4.

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