A175577 -id:A175577 - cp's OEIS frontend

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

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

A005900 Octahedral numbers: a(n) = n(2n^2 + 1)/3.

Original entry on oeis.org

0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680

Offset: 0

N. J. A. Sloane

Series reversion of g.f.: A(x) is Sum_{n>0} - A066357(n)(-x)^n.
Partial sums of centered square numbers A001844. - Paul Barry, Jun 26 2003
Also as a(n) = (1/6)*(4n^3 + 2n), n>0: structured tetragonal diamond numbers (vertex structure 5) (cf. A000447 - structured diamonds); and structured trigonal anti-prism numbers (vertex structure 5) (cf. A100185 - structured anti-prisms). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Schlaefli symbol for this polyhedron: {3,4}.
If X is an n-set and Y and Z are disjoint 2-subsets of X then a(n-4) is equal to the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Starting with 1 = binomial transform of [1, 5, 8, 4, 0, 0, 0, ...] where (1, 5, 8, 4) = row 3 of the Chebyshev triangle A081277. - Gary W. Adamson, Jul 19 2008
a(n) = largest coefficient of (1 + ... + x^(n-1))^4. - R. H. Hardin, Jul 23 2009
Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...) = A005408(x). - Gary W. Adamson, Jul 27 2009
Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4, ...]. - Gary W. Adamson, Jul 28 2009
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral, and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
Let b be any product of four different primes. Then the divisor lattice of b^n is of width a(n+1). - Jean Drabbe, Oct 13 2010
Arises in Bezdek's proof on contact numbers for congruent sphere packings (see preprint). - Jonathan Vos Post, Feb 08 2011
Euler transform of length 2 sequence [6, -2]. - Michael Somos, Mar 27 2011
a(n+1) is the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 2n. - Clark Kimberling, Mar 19 2012
a(n) is the number of semistandard Young tableaux over all partitions of 3 with maximal element <= n. - Alois P. Heinz, Mar 22 2012
Self convolution of the odd numbers. - Reinhard Zumkeller, Apr 04 2012
a(n) is the number of (w,x,y,z) with all terms in {1,...,n} and w+x=y+z; also the number of (w,x,y,z) with all terms in {0,...,n} and |w-x|<=y. - Clark Kimberling, Jun 02 2012
The sequence is the third partial sum of (0, 1, 3, 4, 4, 4, ...). - Gary W. Adamson, Sep 11 2015
a(n) is the number of join-irreducible elements in the Weyl group of type B_n with respect to the strong Bruhat order. - Rafael Mrden, Aug 26 2020
Number of unit octahedra contained in an n-scale octahedron composed of a tetrahedral-octahedral honeycomb. The number of unit tetrahedra in it is 8*A000292(n-1) = 4*(n^3 - n)/3. Also, the number of unit tetrahedra and unit octahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb is respectively A006527(n) = (n^3 + 2*n)/3 and A000292(n-1) = (n^3 - n)/6. - Jianing Song, Feb 24 2025

			G.f. = x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Karoly Bezdek, Contact numbers for congruent sphere packings, arXiv:1102.1198 [math.MG], 2011.
Matteo Cavaleri and Alfredo Donno, Some degree and distance-based invariants of wreath products of graphs, arXiv:1805.08989 [math.CO], 2018.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (5), m=2.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Muhammad Fatih Killik and Bünyamin Şahi̇n, Further Results on Level Matrix, Int'l J. Math. Combin. (2024) Vol 4, 68-74. See p. 71.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrđen, Join operation for the Bruhat order and Verma modules, arXiv:2109.01067 [math.RT], 2021. See Remark 5.10 p. 19.
A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), #R27.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
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
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Octahedral Number.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Sums of 2 consecutive terms give A001845. Cf. A001844.
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.
Cf. A022521.
Cf. A081277.
Row n=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A005408.
Cf. A053676, A053677, A053678.
Cf. A002061.
Cf. A000292 (tetrahedral numbers), A000578 (cubes), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Similar sequence: A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193(m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).

a005900 n = sum $ zipWith (*) odds $ reverse odds
            where odds = take n a005408_list
a005900_list = scanl (+) 0 a001844_list
-- Reinhard Zumkeller, Jun 16 2013, Apr 04 2012

[n*(2*n^2+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015

I:=[0,1,6,19]; [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..50]]; // Vincenzo Librandi, Sep 12 2015

al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(3,n), n=0..100)];
A005900:=(z+1)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(4,2*n)/12, n=0..40); # Zerinvary Lajos, Apr 21 2008

Table[(2n^3+n)/3, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,6,19},50] (* Harvey P. Dale, Oct 10 2013 *)
CoefficientList[Series[x (1 + x)^2/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)

makelist(n*(2*n^2+1)/3, n, 0, 20); /* Martin Ettl, Jan 07 2013 */

PARI
```
{a(n) = n*(2*n^2+1)/3};
```

concat([0],Vec(x*(1 + x)^2/(1 - x)^4 + O(x^50))) \\ Indranil Ghosh, Mar 16 2017

def a(n): return n*(2*n*n + 1)//3
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 03 2021

a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. - Amarnath Murthy, May 28 2001
G.f.: x * (1 + x)^2 / (1 - x)^4. a(n) = -a(-n) = (2*n^3 + n) / 3.
a(n) = ( ((n+1)^5-n^5) - (n^5-(n-1)^5) )/30. - Xavier Acloque, Oct 17 2003
a(n) is the sum of the products pq, where p and q are both positive and odd and p + q = 2n, e.g., a(4) = 7*1 + 5*3 + 3*5 + 1*7 = 44. - Jon Perry, May 17 2005
a(n) = 4*binomial(n,3) + 4*binomial(n,2) + binomial(n,1). - Mitch Harris, Jul 06 2006
a(n) = binomial(n+2,3) + 2*binomial(n+1,3) + binomial(n,3), (this pair generalizes; see A014820, the 4-cross polytope numbers).
Sum_{n>=1} 1/a(n) = 3*gamma + 3*Psi((I*(1/2))*sqrt(2)) - (1/2)*(3*I)*Pi*coth((1/2)*Pi*sqrt(2)) - (1/2)*(3*I)*sqrt(2) = A175577, where I=sqrt(-1). - Stephen Crowley, Jul 14 2009
a(n) = A035597(n)/2. - J. M. Bergot, Jun 11 2012
a(n) = A000578(n) - 2*A000292(n-1) for n>0. - J. M. Bergot, Apr 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Wesley Ivan Hurt, Sep 11 2015
E.g.f.: (1/3)*x*(3 + 6*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Mar 16 2017
a(n) = (A002061(A002061(n+1)) - A002061(A002061(n)))/6. - Daniel Poveda Parrilla, Jun 10 2017
a(n) = 6*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
Sum_{n >= 1} (-1)^(n+1)/(a(n)*a(n+1)) = 6*log(2) - 4 = 1/(6 + 2/(6 + 6/(6 + ... + n*(n-1)/(6 + ...)))). See A142983. - Peter Bala, Mar 06 2024

Sum of the inverses of the tetrahedral numbers (A000292). - Michael B. Porter, Nov 27 2017

For any triangle ABC, cos A + cos B + cos C <= 3/2; equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 17 2022

Defined by sum_{n>=1} 1/A006564(n) = 1/1 + 1/12 +1/48 + 1/124 +...

Equals gamma + Pi*sqrt(5/3)*tanh(Pi*sqrt(15)/10)/2 + Re psi( 1/2+i*sqrt(15)/10 ), where psi is the digamma function, i the imaginary unit, Pi = A000796, sqrt(15)=A010472, gamma=A001620.

cp's OEIS Frontend

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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A005900 Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.

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A152623 Decimal expansion of 3/2.

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A175578 Decimal expansion of the sum over the inverse icosahedral numbers.

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A295421 Decimal expansion of the sum of the reciprocals of the dodecahedral numbers (A006566).

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A334926 G.f.: Sum_{k>=1} x^(k*(2*k^2 + 1)/3) / (1 - x^(k*(2*k^2 + 1)/3)).

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A005900 Octahedral numbers: a(n) = n(2n^2 + 1)/3.

A334926 G.f.: Sum_{k>=1} x^(k(2k^2 + 1)/3) / (1 - x^(k(2k^2 + 1)/3)).