A000796 -id:A000796 - cp's OEIS frontend

A062964 Pi in hexadecimal.

3, 2, 4, 3, 15, 6, 10, 8, 8, 8, 5, 10, 3, 0, 8, 13, 3, 1, 3, 1, 9, 8, 10, 2, 14, 0, 3, 7, 0, 7, 3, 4, 4, 10, 4, 0, 9, 3, 8, 2, 2, 2, 9, 9, 15, 3, 1, 13, 0, 0, 8, 2, 14, 15, 10, 9, 8, 14, 12, 4, 14, 6, 12, 8, 9, 4, 5, 2, 8, 2, 1, 14, 6, 3, 8, 13, 0, 1, 3, 7, 7, 11, 14, 5, 4, 6, 6, 12, 15, 3, 4, 14, 9

Offset: 1

Robert Lozyniak (11(AT)onna.com), Jul 22 2001

Bailey and Crandall conjecture that the terms of this sequence, apart from the first, are given by the formula floor(16*(x(n) - floor(x(n)))), where x(n) is determined by the recurrence equation x(n) = 16*x(n-1) + (120*n^2 - 89*n + 16)/(512*n^4 - 1024*n^3 + 712*n^2 - 206*n + 21) with the initial condition x(0) = 0 (see A374334). They have numerically verified the conjecture for the first 100000 terms of the sequence. - Peter Bala, Oct 31 2013

Bailey, Borwein & Plouffe's ("BBP") formula allows one to compute the n-th hexadecimal digit of Pi without calculating the preceding digits (see Wikipedia link). - M. F. Hasler, Mar 14 2015

			3.243f6a8885a308d3...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 17-28.

Harry J. Smith, Table of n, a(n) for n = 1..20000
D. H. Bailey, Compendium of BBP-Type Formulas for Mathematical Constants.
D. H. Bailey and R. E. Crandall, On the Random Character of Fundamental Constant Expansions, Experiment. Math. Volume 10, Issue 2 (2001), 175-190.
CalcCrypto, Pi in Hexadecimal. [Broken link]
Steven R. Finch, The Miraculous Bailey-Borwein-Plouffe Pi Algorithm.
Steve Pagliarulo, Stu's pi page: base 16 (31 pages of numbers). [Dead link]
Johnny Vogler, More digits.
Wikipedia, Bailey-Borwein-Plouffe formula.

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), A000796 (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), this sequence (b=16), A060707 (b=60).

Cf. A007514, A374334.

Mathematica
```
RealDigits[ N[ Pi, 115], 16] [[1]]
```

{ default(realprecision, 24300); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*16; write("b062964.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009

N=50; default(realprecision,.75*N); A062964=digits(Pi*16^N\1,16) \\ M. F. Hasler, Mar 14 2015

a(n) = 8*A004601(4n) + 4*A004601(4n+1) + 2*A004601(4n+2) + 1*A004601(4n+3).

If Pi is the expansion of Pi in base 10, Pi=3.1415926...: a(n) = floor(16^n*Pi) - 16*floor(16^(n-1)*Pi). - Benoit Cloitre, Mar 09 2002

More terms from Henry Bottomley, Jul 24 2001

A019609 Decimal expansion of Pi*e.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Not known to be irrational (though of course conjectured transcendental), see e.g. Klee & Wagon. - Charles R Greathouse IV, Jul 23 2015

			8.53973422267356706546355086954657449503488853576511496187960113...

Victor Klee and Stan Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Mathematical Association of America (1991). Problem 22, p. 243.

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A159822 (continued fraction for Pi*e).

Cf. also A000796 (Pi), A001113 (e).

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)*Exp(1); // G. C. Greubel, Aug 24 2018

RealDigits[N[Pi*E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

default(realprecision, 20080); x=Pi*exp(1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019609.txt", n, " ", d)); \\ Harry J. Smith, Apr 27 2009

Limit_{k->oo} 4k/u(k)^2 where u(1)=0, u(2)=1, u(k+2) = u(k+1) + u(k)/(2k). - Benoit Cloitre, Aug 14 2003

Equals Product_{k>=0} ((k + 1)^(4*k + 3)/(k + 2)^(6*k + 5))*(((2*k + 3)*(k + 3))/(2*k + 1))^(2*k + 2). - Antonio Graciá Llorente, May 31 2024

Checked by Neven Juric (neven.juric(AT)apis-it.hr), Feb 04 2008

A019727 Decimal expansion of sqrt(2*Pi).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Pickover says that the expression: lim_{n->oo} e^n(n!) / (n^n * sqrt(n)) = sqrt(2*Pi) is beautiful because it connects Pi, e, radicals, factorials and infinite limits. - Jason Earls, Mar 16 2001
Appears in the formula of the normal distribution. - Johannes W. Meijer, Feb 23 2013
sqrt(2*Pi)*sqrt(n) is the expected height of a labeled random tree of order n (see Rényi, Szekeres, 1967, formula (4.6)). - Hugo Pfoertner, May 18 2023
The constant in the formula known as "Stirling's approximation" (or "Stirling's formula"). It is sometimes called Stirling constant. The formula without the exact value of the constant was discovered by the French mathematician Abraham de Moivre (1667-1754), and was published in his book (1730). The exact value of the constant was found by the Scottish mathematician James Stirling (1692-1770) and was published in his book "Methodus differentialis" (1730). - Amiram Eldar, Jul 08 2023

			2.506628274631000502415765284811045253006986740609938316629923576342293....

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.
Clifford A. Pickover, Wonders of Numbers, Oxford University Press, NY, 2001, p. 307.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
Abraham de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, London, England: J. Tonson & J. Watts, 1730, pp. 96-106.
K. Kimoto, N. Kurokawa, C. Sonoki, and M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
A. Rényi and G. Szekeres, On the height of trees, Journal of the Australian Mathematical Society , Volume 7 , Issue 4 , November 1967 , pp. 497-507.
James Stirling, Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum, London, 1730. See Propositio XXVIII, pp. 135-139.
Eric Weisstein's World of Mathematics, Normal Distribution.
Wikipedia, Stirling's approximation.
Index entries for transcendental numbers.

Cf. A058293 (continued fraction), A231863 (inverse), A000796 (Pi).

R:= RealField(100); Sqrt(2*Pi(R)); // G. C. Greubel, Mar 08 2018

RealDigits[Sqrt[2Pi],10,120][[1]] (* Harvey P. Dale, Dec 12 2012 *)

fpprec: 100$ ev(bfloat(sqrt(2*%pi))); /* Martin Ettl, Oct 11 2012 */

default(realprecision, 20080); x=sqrt(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019727.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

Equals lim_{n->oo} e^n*(n!)/n^n*sqrt(n).
Also equals Integral_{x >= 0} W(1/x^2) where W is the Lambert function, which is also known as ProductLog. - Jean-François Alcover, May 27 2013
Also equals the generalized Glaisher-Kinkelin constant A_0, see the Finch reference. - Jean-François Alcover, Dec 23 2014
Equals exp(-zeta'(0)). See Kimoto et al. - Michel Marcus, Jun 27 2019

A079586 Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 26 2003

André-Jeannin proved that this constant is irrational.

This constant does not belong to the quadratic number field Q(sqrt(5)) (Bundschuh and Väänänen, 1994). - Amiram Eldar, Oct 30 2020

			3.35988566624317755317201130291892717968890513373...

Daniel Duverney, Number Theory, World Scientific, 2010, 5.22, pp.75-76.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.

Kenny Lau, Table of n, a(n) for n = 1..10000 (First 1000 terms computed by Joerg Arndt)
Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541.
Richard André-Jeannin, Problem H-450, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 1 (1991), p. 89; Comparable, Solution to Problem H-450 by Paul S. Bruckman, ibid., Vol. 30, No. 2 (1992), p. 191-192.
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208;
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Paul S. Bruckman, Problem B-602, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 25, No. 3 (1987), p. 279; Fibonacci Infinite Series, Solution to Problem B-602 by C. Georghiou, ibid., Vol. 26, No. 3 (1988), pp. 281-282.
Peter Bundschuh and Keijo Väänänen, Arithmetical investigations of a certain infinite product, Compositio Mathematica, Vol. 91, No. 2 (1994), pp. 175-199.
Daniel Duverney, Irrationalité de la somme des inverses de la suite Fibonacci, Elemente der Mathematik, Vol. 52, No. 1 (1997), pp. 31-36.
William Gosper, Acceleration of Series, Artificial Intelligence Memo #304 (1974).
W. E. Greig, Sums of Fibonacci reciprocals, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 46-48.
Peter Griffin, Acceleration of the Sum of Fibonacci Reciprocals, The Fibonacci Quarterly, Vol. 30, No. 2 (1992), pp. 179-181.
Sarah H. Holliday and Takao Komatsu, On the sum of reciprocal generalized Fibonacci numbers, Integers 11A (2011), Article 11. Alternate link.
A. F. Horadam, Elliptic functions and Lambert series in the summation of reciprocals in certain recurrence-generated sequences, The Fibonacci Quarterly, Vol. 26, No.2 (May-1988), pp. 98-114.
Fredrik Johansson, The reciprocal Fibonacci constant.
Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021).
Tapani Matala-Aho and Marc Prévost, Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers, Ramanujan J., Vol. 11 (2006), pp. 249-261.
Z.W. M. Trzaska, Fibonacci Polynomials their Properties and Applications, Z. Anal. Anwend. 15 (1996), no. 3, pp. 729-746.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Wikipedia, Reciprocal Fibonacci constant.

Cf. A000045, A000796, A001622, A002163, A002390, A084119, A093540, A016628, A105199, A153386, A153387.

Cf. A350901, A350902, A350903, A350904.

Digits := 120: c := Pi/2 + I*arccsch(2):
Jeannin := n -> sqrt(5/4)*add(I^(1-j)/sin(j*c), j = 1..n):
evalf(Jeannin(1000)); # Peter Luschny, Nov 15 2023

digits = 105; Sqrt[5]*NSum[(-1)^n/(GoldenRatio^(2*n + 1) - (-1)^n), {n, 0, Infinity}, WorkingPrecision -> digits, NSumTerms -> digits] // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Apr 09 2013 *)
First@RealDigits[Sqrt[5]/4 ((Log[5] + 2 QPolyGamma[1, 1/GoldenRatio^4] - 4 QPolyGamma[1, 1/GoldenRatio^2])/(2 Log[GoldenRatio]) + EllipticTheta[2, 0, 1/GoldenRatio^2]^2), 10, 105] (* Vladimir Reshetnikov, Nov 18 2015 *)

/* Fast computation without splitting into even and odd indices, see the Arndt reference */
lambert2(x, a, S)=
{
/* Return G(x,a) = Sum_{n>=1} a*x^n/(1-a*x^n) (generalized Lambert series)
   computed as Sum_{n=1..S} x^(n^2)*a^n*( 1/(1-x^n) + a*x^n/(1-a*x^n) )
   As series in x correct up to order S^2.
   We also have G(x,a) = Sum_{n>=1} a^n*x^n/(1-x^n) */
    return( sum(n=1,S, x^(n^2)*a^n*( 1/(1-x^n) + a*x^n/(1-a*x^n) ) ) );
}
inv_fib_sum(p=1, q=1, S)=
{
/* Return Sum_{n>=1} 1/f(n) where f(0)=0, f(1)=1, f(n) = p*f(n-1) + q*f(n-1)
   computed using generalized Lambert series.
   Must have p^2+4*q > 0 */
    my(al,be);
    \\ Note: the q here is -q in the Horadam paper.
    \\ The following numerical examples are for p=q=1:
    al=1/2*(p+sqrt(p^2+4*q));  \\ == +1.6180339887498...
    be=1/2*(p-sqrt(p^2+4*q));  \\ == -0.6180339887498...
    return( (al-be)*( 1/(al-1) + lambert2(be/al, 1/al, S) ) ); \\ == 3.3598856...
}
default(realprecision,100);
S = 1000; /* (be/al)^S == -0.381966^S == -1.05856*10^418 << 10^-100 */
inv_fib_sum(1,1,S) /* 3.3598856... */ /* Joerg Arndt, Jan 30 2011 */

suminf(k=1, 1/(fibonacci(k))) \\ Michel Marcus, Feb 19 2019

m=120; numerical_approx(sum(1/fibonacci(k) for k in (1..10*m)), digits=m) # G. C. Greubel, Feb 20 2019

Alternating series representation: 3 + Sum_{k >= 1} (-1)^(k+1)/(F(k)*F(k+1)*F(k+2)). - Peter Bala, Nov 30 2013
From Amiram Eldar, Oct 04 2020: (Start)
Equals sqrt(5) * Sum_{k>=0} (1/(phi^(2*k+1) - 1) - 2*phi^(2*k+1)/(phi^(4*(2*k+1)) - 1)), where phi is the golden ratio (A001622) (Greig, 1977).
Equals sqrt(5) * Sum_{k>=0} (-1)^k/(phi^(2*k+1) - (-1)^k) (Griffin, 1992).
Equals A153386 + A153387. (End)
From Gleb Koloskov, Sep 14 2021: (Start)
Equals 1 + c1*(c2 + 32*Integral_{x=0..infinity} f(x) dx),
where c1 = sqrt(5)/(8*log(phi)) = A002163/(8*A002390),
      c2 = 2*arctan(2)+log(5) = 2*A105199+A016628,
      phi = (1+sqrt(5))/2 = A001622,
f(x) = sin(x)*(4+cos(2*x))/((exp(Pi*x/log(phi))-1)*(2*cos(2*x)+3)*(7-2*cos(2*x))) (End)
From Amiram Eldar, Jan 27 2022: (Start)
Equals 3 + 2 * Sum_{k>=1} 1/(F(2*k-1)*F(2*k+1)*F(2*k+2)) (Bruckman, 1987).
Equals 2 + Sum_{k>=1} 1/A350901(k) (André-Jeannin, Problem H-450, 1991).
Equals lim_{n->oo} A350903(n)/(A350904(n)*A350902(n)) (André-Jeannin, 1991). (End)
Equals sqrt(5/4)*Sum_{j>=1} i^(1-j)/sin(j*c) where c = Pi/2 + i*arccsch(2). - Peter Luschny, Nov 15 2023
Equals lim_{n->oo} A203006(n)/A003266(n) (Z.W. M. Trzaska, 1996). - Raul Prisacariu, Sep 04 2024

A093766 Decimal expansion of Pi/(2*sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - R. J. Mathar, Nov 22 2011
Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - Omar E. Pol, Dec 09 2013

			0.906899682117108925297039128821077866142033124046370287784942...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.

J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
Xi Lin, Dirk Schmelter, Sadaf Imanian, and Horst Hintze-Bruening, Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid, Scientific Reports (2019) Vol. 9, Article No. 16389.
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. See Table 22 for L(m=6,r=2,s=1).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 625-626.
N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
Eckard Specht, May 21 2012, The best known packings of equal circles in a circle (complete up to N=1500).
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
Eric Weisstein's World of Mathematics, Smoothed Octagon.
Eric Weisstein's World of Mathematics, Circle Packing.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.

Cf. A072691, A381671.

RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)

Pi/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014

Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
Equals Sum_{n>=1} 1/A134667(n). [Jolley]
Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
Equals A000796 / A010469. - Omar E. Pol, Dec 09 2013
Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - Peter Bala, Feb 04 2015
From Peter Bala, Feb 16 2015: (Start)
Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... =  1.102657790843585... - Dimitris Valianatos, Aug 31 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021
For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024
From Stefano Spezia, Jun 05 2025: (Start)
Equals Sum_{k>=0} (-1)^k/((k + 1)*(3*k + 1)).
Equals Integral_{x=0..oo} 1/(x^4 + x^2 + 1) dx.
Equals Integral_{x=0..oo} x^2/(x^4 + x^2 + 1) dx. (End)
Equals sqrt(A072691) = 3*A381671. - Hugo Pfoertner, Jun 05 2025

Entry revised by N. J. A. Sloane, Feb 10 2013

A091925 Decimal expansion of Pi^3.

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Mar 16 2004

Surface area of the 6-dimensional unit sphere. - Stanislav Sykora, Nov 08 2013
Surface area of a sphere of diameter Pi equals the volume of the circumscribed cube. - Omar E. Pol, Dec 25 2013
Area of a circle of radius Pi. - Omar E. Pol, Jan 31 2016

			31.00627668029982017547631506710139520222528856588510769414453810380639...

Harry J. Smith, Table of n, a(n) for n = 2..20000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 31
Jean-Christophe Pain, Series representations for Pi^3 involving the golden ratio, arXiv:2206.15281 [math.NT], 2022.
Khodabakhsh Hessami Pilehrood, Tatiana Hessami Pilehrood, Series acceleration formulas for beta values, Discrete Mathematics & Theoretical Computer Science 12:2 (2010), pp. 223-236.
Stanislav Sykora, Surface Integrals over n-Dimensional Spheres.
Index entries for transcendental numbers

Cf. A000796, A002388, A058285 (continued fraction), A019670, A093954, A092731, A092735.

R:= RealField(100); (Pi(R))^3; // G. C. Greubel, Mar 09 2018

First@ RealDigits@ N[Pi^3, 120] (* Michael De Vlieger, Jan 31 2016 *)

default(realprecision, 20080); x=Pi^3/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b091925.txt", n, " ", d)); \\ Harry J. Smith, Jun 22 2009

Sum_{k >= 0} binomial(2*k,k)/((2*k + 1)^3*16^k) = 7*Pi^3/216. (Kh. Hessami Pilehrood and T. Hessami Pilehrood).
From Peter Bala, Feb 05 2015: (Start)
The integer sequences A(n) := 2^n*(2*n + 1)!^3/n!^2 and B(n) := A(n)*( Sum {k = 0..n} binomial(2*k,k)*1/(2*k + 1)^3*(1/16)^k ) both satisfy the second order recurrence equation u(n) = (160*n^4 + 128*n^3 + 144*n^2 + 2)*u(n-1) - 32*(n - 1)*(2*n - 1)^7*u(n-2). From this observation we can obtain the continued fraction expansion 7/216*Pi^3 = 1 + 2/(432 - 32*3^7/(4162 - 32*2*5^7/(17714 - ... - 32*(n - 1)*(2*n - 1)^7/((160*n^4 + 128*n^3 + 144*n^2 + 2) - ... )))). Cf. A002388, A019670 and A093954. (End)
From Peter Bala, Oct 31 2019: (Start)
Pi^3 = (1/7) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/6)^3 + 1/(n + 5/6)^3 ).
Pi^3 = (1/31) * Sum_{n >= 0} (-1)^n*( 1/(n + 1/10)^3 - 1/(n + 3/10)^3 - 1/(n + 7/10)^3 + 1/(n + 9/10)^3 ). Cf. A019692, A092731 and A092735. (End)
Equals Integral_{x=-oo..oo} x^2/(exp(x/2) + exp(-x/2)) dx. - Amiram Eldar, May 21 2021

A037000 Positions of the digit '1' in the decimal expansion of Pi.

Original entry on oeis.org

1, 3, 37, 40, 49, 68, 94, 95, 103, 110, 138, 148, 153, 154, 155, 163, 168, 174, 175, 198, 206, 220, 238, 243, 246, 250, 269, 281, 295, 297, 314, 319, 324, 342, 344, 362, 363, 381, 385, 390, 393, 395, 396, 417, 424, 427, 428, 432, 437, 438, 442, 445, 446

Offset: 1

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

From M. F. Hasler, Jul 28 2024: (Start)
"Positions" are indices n of digits d(n) such that Pi = Sum_{n >= 0} d(n)/10^n; see A053745 for the variant where the initial digit 3 is at position 1.
The first few primes in this sequence are 3, 37, 103, 163, 269, 281, 499, 541, 547, 587, 607, 709, 797, 859, 887, 971, 983, 997, ... (End)

Robert Israel, Table of n, a(n) for n = 1..10137
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796 (decimals of Pi), A037001 - A037008 and A036974 (positions of other digits), A053745 (variant with all values increased by 1).

P:= convert(evalf[100000](Pi),string)[3..-1]:
select(t -> P[t]="1",[$1..length(P)-1]); # Robert Israel, Dec 22 2013

Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 1] (* Robert G. Wilson v, Mar 07 2011 *)

A037000_upto(N=500, d=1)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024

Conjecturally, a(n) ~ 10n.

A068465 Decimal expansion of Gamma(3/4).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 10 2002

			Gamma(3/4) = 1.225416702465177645129098303362890526851239248108070611...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:14 at page 414.

G. C. Greubel, Table of n, a(n) for n = 1..20000
Russell J. Matheson, GAMMA(3/4) computed to 14550 digits.
Simon Plouffe, GAMMA(3/4) to 256 places, see p. 65.
Index to sequences related to the Gamma function

Cf. A000796, A002193, A053004, A063448, A068466, A069998.

SetDefaultRealField(RealField(105)); Gamma(3/4); // G. C. Greubel, Mar 11 2018

evalf(GAMMA(3/4)) ; # R. J. Mathar, Jan 10 2013

RealDigits[Gamma[3/4], 10, 100][[1]] (* G. C. Greubel, Mar 11 2018 *)

default(realprecision, 100); gamma(3/4) \\ G. C. Greubel, Mar 11 2018

Gamma(3/4) * A068466 = sqrt(2)*Pi = A063448. - R. J. Mathar, Jun 18 2006
Equals Integral_{x>=0} x^(-1/4)*exp(-x) dx. - Vaclav Kotesovec, Nov 12 2020
Equals (Pi/2)^(1/4) * sqrt(AGM(1,sqrt(2))) = sqrt(A069998 * A053004). - Amiram Eldar, Jun 12 2021

A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jun 25 2001

			2.622057554292119810464839589891119413682754951431623162816821703...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.3 and 6.2, pp. 99, 420.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits).
Simon Plouffe, Lemniscate or Gauss constant.
Simon Plouffe, Lemniscate constant or Gauss constant.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Lemniscate constant.
Index entries for transcendental numbers.

Cf. A062540, A064853, A085565.
Cf. A002193, A014549, A093341.
Equals A000796/A053004 (see PARI script).

SetDefaultRealField(RealField(100)); R:= RealField(); (1/2)*Sqrt(2*Pi(R)^3)/Gamma(3/4)^2; // G. C. Greubel, Oct 07 2018

evalf((1/2)*sqrt(2*Pi^3)/GAMMA(3/4)^2,120); # Muniru A Asiru, Oct 08 2018
evalf(1/2*GAMMA(1/4)*GAMMA(1/2)/GAMMA(3/4),120); # Martin Renner, Aug 16 2019
evalf(1/2*Beta(1/4,1/2),120); # Martin Renner, Aug 16 2019
evalf(2*int(1/sqrt(1-x^4),x=0..1),120); # Martin Renner, Aug 16 2019

RealDigits[Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2, 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

print(1/2*Pi^(3/2)/gamma(3/4)^2*2^(1/2))

allocatemem(932245000); default(realprecision, 5080); x=Pi^(3/2)*sqrt(2)/(2*gamma(3/4)^2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062539.txt", n, " ", d)); \\ Harry J. Smith, Jun 20 2009

Pi/agm(1,sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2015

intnum(x=0,Pi, 1/sqrt(1 + sin(x)^2)) \\ Charles R Greathouse IV, Feb 04 2025

Equals (1/2)*sqrt(2*Pi^3)/Gamma(3/4)^2.
A093341 multiplied by A002193. - R. J. Mathar, Aug 28 2013
From Martin Renner, Aug 16 2019: (Start)
Equals 2*Integral_{x=0..1} 1/sqrt(1-x^4) dx.
Equals 1/2*B(1/4,1/2) with Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y). (End)
Equals Pi/AGM(1, sqrt(2)). - Jean-François Alcover, Feb 28 2021
Equals 2*hypergeom([1/2, 1/4], [5/4], 1). - Peter Bala, Mar 02 2022
Equals (1/2)*A064853 = 2*A085565. - Amiram Eldar, May 04 2022
Equals Pi*A014549. - Hugo Pfoertner, Jun 28 2024
Equals Integral_{x=0..Pi} 1/sqrt(1 + sin(x)^2) dx = EllipticK(-1) (see Finch at p. 420). - Stefano Spezia, Dec 15 2024
Equals Gamma(1/4)^2 / (sqrt(Pi)*2^(3/2)). - Vaclav Kotesovec, Apr 26 2025
Equals (161*6440^(1/4))/(2*Sum_{k>=0} N(k)/D(k)) with N(k) = Pochhammer(1/8,k) * Pochhammer(5/8,k) * (275+8640*k) and D(k) = (k!)^2*25921^k [Jorge Zuniga, 2023].

A007514 Pi = Sum_{n >= 0} a(n)/n!.

Original entry on oeis.org

3, 0, 0, 0, 3, 1, 5, 6, 5, 0, 1, 4, 7, 8, 0, 6, 7, 10, 7, 10, 4, 10, 6, 16, 1, 11, 20, 3, 18, 12, 9, 13, 18, 21, 14, 34, 27, 11, 27, 33, 36, 18, 5, 18, 5, 23, 39, 1, 10, 42, 28, 17, 20, 51, 8, 42, 47, 0, 27, 23, 16, 52, 32, 52, 53, 24, 43, 61, 64, 18, 17, 11, 0, 53, 14, 62

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

The current name does not define a(n) without ambiguity. It is meant that for each n, a(n) is the largest integer such that the remainder of Pi - (partial sum up to n) remains positive. This leads to the FORMULA given below. - M. F. Hasler, Mar 20 2017

			Pi = 3/0! + 0/1! + 0/2! + 0/3! + 3/4! + 1/5! + ...

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

Hans Havermann, Table of n, a(n) for n = 0..10000
Index entries for sequences related to the number Pi

Essentially same as A075874.

Pi in base n: A004601 to A004608, A000796, A068436 to A068440, A062964.

p = N[Pi, 1000]; Do[k = Floor[p*n! ]; p = p - k/n!; Print[k], {n, 0, 75} ]

x=Pi;vector(floor((y->y/log(y))(default(realprecision))),n,t=(n-1)!;k=floor(x*t);x-=k/t;k) \\ Charles R Greathouse IV, Jul 15 2011

C=1/Pi;x=0;vector(primepi(default(realprecision)),n,-x*n--+x=n!\C) \\ M. F. Hasler, Mar 20 2017

a(n) = floor(n!*Pi) - n*floor((n-1)!*Pi) for all n > 0. - M. F. Hasler, Mar 20 2017

cp's OEIS Frontend

A062964 Pi in hexadecimal.

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A019609 Decimal expansion of Pi*e.

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A019727 Decimal expansion of sqrt(2*Pi).

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A079586 Decimal expansion of Sum_{k>=1} 1/F(k) where F(k) is the k-th Fibonacci number A000045(k).

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A093766 Decimal expansion of Pi/(2*sqrt(3)).

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A091925 Decimal expansion of Pi^3.

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