cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 81-90 of 1164 results. Next

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

Original entry on oeis.org

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

Views

Author

Bruno Berselli, Mar 06 2015

Keywords

Comments

Value of the Borwein-Borwein function I_3(a,b) for a = b = 1. - Stanislav Sykora, Apr 16 2015
The area of a circle circumscribing a unit-area regular hexagon. - Amiram Eldar, Nov 05 2020

Examples

			1.2091995761561452337293855050947704881893774987284937170465899569254...
		

References

  • George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), pp. 120-121.
  • L. B. W. Jolley, Summation of Series, Dover (1961), No. 261, pp. 48, 49, (and No. 275).

Crossrefs

Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).

Programs

  • Mathematica
    RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]
  • PARI
    a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015

Formula

Equals 2*sqrt(3)*Pi/9 = 1 + 1/6 + 1/30 + 1/140 + 1/630 + 1/2772 + 1/12012 + ...
Equals m*I_3(m,m) = m*Integral_{x>=0} (x/(m^3+x^3)), for any m>0. - Stanislav Sykora, Apr 16 2015
Equals Integral_{x>=0} (1/(1+x^3)) dx. - Robert FERREOL, Dec 23 2016
From Peter Bala, Oct 27 2019: (Start)
Equals 3/4*Sum_{n >= 0} (n+1)!*(n+2)!/(2*n+3)!.
Equals Sum_{n >= 1} 3^(n-1)/(n*binomial(2*n,n)).
Equals 2*Sum_{n >= 1} 1/(n*binomial(2*n,n)). See Boros and Moll, pp. 120-121.
Equals Integral_{x = 0..1} 1/(1 - x^3)^(1/3) dx = Sum_{n >= 0} (-1)^n*binomial(-1/3,n) /(3*n + 1).
Equals 2*Sum_{n >= 1} 1/((3*n-1)*(3*n-2)) = 2*(1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...) (added Oct 30 2019). (End)
Equals Product_{k>=1} 9*k^2/(9*k^2 - 1). - Amiram Eldar, Aug 04 2020
From Peter Bala, Dec 13 2021: (Start)
Equals (2/3)*A093602.
Conjecture: for k >= 0, 2*sqrt(3)*Pi/9 = (3/2)^k * k!*Sum_{n = -oo..oo} (-1)^n/ Product_{j = 0..k} (3*n + 3*j + 1). (End)
Equals (3/4)*S - 1, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..Pi/2} tan(x)^(1/3)/(sin(2*x) + 1) dx. See MIT Link. - Joost de Winter, Aug 26 2023
Continued fraction: 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). See A000407. - Peter Bala, Feb 20 2024
Equals Sum_{n>=2} 1/binomial(n, floor(n/2)); and trivially if "floor" is replaced by "ceiling". - Richard R. Forberg, Aug 30 2024
Equals Product_{k>=2} (1 + (-1)^k/A001651(k)). - Amiram Eldar, Nov 22 2024
Equals 2*A073010 = 1/A086089 = sqrt(A214549) = exp(A256923) = A275486/2. - Hugo Pfoertner, Nov 22 2024
Equals 1 - (1/6) * Sum_{n>=1} A010815(n)/n. - Friedjof Tellkamp, Apr 05 2025
Equals A248181 - 2. - Pontus von Brömssen, Apr 05 2025

A381684 Decimal expansion of the isoperimetric quotient of a truncated tetrahedron.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Mar 04 2025

Keywords

Comments

Polya (1954) defines the isoperimetric quotient of a solid as 36*Pi*V^2/S^3, where V and S are the volume and surface area of the solid, respectively.
The isoperimetric quotient of a sphere is 1.

Examples

			0.4662292826432950646084875599089894958106273300491...
		

References

  • George Polya, Mathematics and Plausible Reasoning, Vol. 1: Induction and Analogy in Mathematics, Princeton University Press, Princeton, New Jersey, 1954. See pp. 188-189, exercise 43.

Crossrefs

Cf. A377274 (surface area), A377275 (volume).

Programs

  • Mathematica
    First[RealDigits[529*Pi/(2058*Sqrt[3]), 10, 100]]
  • PARI
    529*Pi/2058/sqrt(3) \\ Charles R Greathouse IV, Aug 19 2025

Formula

Equals 36*Pi*A377275^2/(A377274^3).
Equals 529*Pi/(2058*sqrt(3)) = 529*A000796/(2058*A002194).

A019675 Decimal expansion of Pi/8.

Original entry on oeis.org

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

Views

Author

Keywords

Comments

Equals Integral_{x>=0} sin(4*x)/(4*x) dx. - Jean-François Alcover, Feb 28 2013
Consider 4 circles inscribed in a square. Inscribe a square in each circle. And finally, inscribe 4 circles inside each four small squares. Totally we get 16 small circles. Pi/8 is the ratio of the area of the 16 small circles to the area of initial square. See the link. - Kirill Ustyantsev, Apr 30 2020

Examples

			Pi/8 = 0.392699081698724154807830422909937860524646174921888227621868... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009
		

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 492.

Crossrefs

Programs

  • Magma
    pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^100*(pi)/8))); // Vincenzo Librandi, Oct 07 2015
  • Mathematica
    RealDigits[N[Pi/8,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)
  • PARI
    default(realprecision, 1002);
    eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(4*n+2)))), "3..-2"))  \\ Gheorghe Coserea, Oct 06 2015
    

Formula

From Peter Bala, Nov 15 2016: (Start)
Pi/8 = Sum_{k >= 1} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)).
More generally, for n >= 0 we have 1/(2*n)! * Pi/4 = Sum_{k >= 1} (-1)^(k+n-1) * 1/Product_{j = -n..n} (2*k + 2*j - 1): when n = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.
For N divisible by 4, we have the asymptotic expansion Pi/8 - Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) ~ -1/2*(1/N^3 - 2/N^5 + 31/N^7 - 692/N^9 + ...), where the sequence of unsigned coefficients [1, 2, 31, 692, ...] equals A024235. (End)
Equals Integral_{x = 0..1} x*sqrt(1 - x^4) dx. - Peter Bala, Oct 27 2019
Equals Integral_{x = 0..oo} sin(x)^6/x^4 dx = Sum_{n >= 1} sin(n)^6/n^4, by the Abel-Plana formula. - Peter Bala, Nov 04 2019
From Amiram Eldar, Jul 12 2020: (Start)
Equals arctan(sqrt(2) - 1).
Equals Sum_{k>=0} (-1)^k/(4*k+2).
Equals Sum_{k>=0} 1/((4*k+1)*(4*k+3)) = Sum_{k>=0} 1/A001539(k).
Equals Integral_{x=0..oo} dx/(x^2 + 16).
Equals Integral_{x=0..oo} dx/(x^4 + 4) = Integral_{x=0..oo} x/(x^4 + 4) dx.
Equals Integral_{x=0..oo} x/(x^4 + 1)^2 dx = Integral_{x=0..1} x/(x^4 + 1) dx.
Equals Integral_{x=0..1} x * arcsin(x) dx. (End)
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals Integral_{x=0..oo} (x*log(x + 1))/((x^2 + 1)^2) dx.
Equals Integral_{x=0..oo} (x^3 - 3*x + 3*arctan(x))/(3*x^5) dx. (End)

A036974 Positions of the digit '7' in decimal expansion of Pi.

Original entry on oeis.org

13, 29, 39, 47, 56, 66, 96, 99, 120, 139, 156, 166, 209, 224, 232, 235, 242, 288, 299, 301, 306, 320, 343, 351, 405, 407, 412, 429, 439, 452, 458, 463, 468, 475, 478, 486, 506, 538, 540, 544, 548, 556, 559, 560, 567, 569, 575, 577, 584, 591, 609, 621, 622
Offset: 1

Views

Author

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

Keywords

Comments

In the first 400 digits of Pi, 7 is rare and only appears 24 times. In the next 250 digits of Pi, 7 is common and appears 38 times. - Bobby Jacobs, Oct 18 2016

Examples

			3.141592653589(7)932384626433832(7)950288419(7)1693993(7)510
58209(7)494459230(7)81640628620899862803482534211(7)06(7)9
8214808651328230664(7)093844609550582231(7)25359408128
48111(7)450284102(7)0193852110555964462294895493038196
44288109(7)56659334461284(7)5648233(7)86(7)831652(7)12019091
4564856692346034861045432664821339360(7)2602491412(7)3
(7)2458(7)0066063155881(7)4881520920962829254091(7)1536436
(7)8925903600113305305488204665213841469519415116094
3305(7)2(7)0365(7)5959195309218611(7)381932611(7)93105118548
0(7)44623(7)9962(7)4956(7)351885(7)52(7)2489122(7)93818301194912
98336(7)3362440656643086021394946395224(7)3(7)190(7)021(7)98
60943(7)02(77)053921(7)1(7)62931(7)6(7)523846(7)481846(7)669405132
00056812(7)14526356082(77)85(77)1342(7)5(77)896091(7)363(7)1(7)8(7)...
		

Crossrefs

Programs

  • Mathematica
    Flatten@Position[RealDigits[Pi - 3, 10, 500][[1]], 7] (* Robert G. Wilson v,Mar 07 2011 *)
    Union[Flatten[SequencePosition[RealDigits[Pi,10,700][[1]],{7}]]]-1 (* Harvey P. Dale, Jul 13 2023 *)

A037002 Positions of the digit '3' in the decimal expansion of Pi - 3.

Original entry on oeis.org

9, 15, 17, 24, 25, 27, 43, 46, 64, 86, 91, 111, 115, 123, 137, 142, 170, 194, 196, 215, 216, 230, 231, 237, 261, 265, 274, 282, 283, 285, 300, 313, 346, 349, 358, 364, 365, 368, 382, 401, 402, 409, 420, 430, 434, 441, 457, 469, 488, 492, 503, 504, 507, 508
Offset: 1

Views

Author

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

Keywords

Crossrefs

Programs

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

A068436 Expansion of Pi in base 11.

Original entry on oeis.org

3, 1, 6, 1, 5, 0, 7, 0, 2, 8, 6, 5, 10, 4, 8, 5, 2, 3, 5, 2, 1, 5, 2, 5, 9, 7, 7, 7, 5, 2, 9, 4, 1, 8, 3, 8, 6, 6, 8, 8, 4, 8, 8, 5, 3, 1, 6, 3, 10, 1, 10, 5, 4, 2, 1, 3, 0, 0, 4, 6, 5, 8, 0, 6, 5, 2, 2, 7, 3, 5, 0, 5, 3, 3, 7, 1, 5, 2, 7, 1, 7, 8, 1, 10, 6, 5, 6, 3, 7, 1, 5, 7, 8, 1, 3, 3, 4, 9, 2, 8, 8, 8
Offset: 1

Views

Author

Benoit Cloitre, Mar 09 2002

Keywords

Examples

			3.16150702865a4...
		

Crossrefs

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), this sequence (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).
Cf. A007514.

Programs

  • Mathematica
    RealDigits[Pi, 11, 111][[1]]
    Table[ResourceFunction["NthDigit"][Pi, n, 11], {n, 1, 111}] (* Joan Ludevid, Oct 09 2022 easy to compute a(10000000)=10 with this function; requires Mathematica 12.0+ *)

A068437 Expansion of Pi in base 12.

Original entry on oeis.org

3, 1, 8, 4, 8, 0, 9, 4, 9, 3, 11, 9, 1, 8, 6, 6, 4, 5, 7, 3, 10, 6, 2, 1, 1, 11, 11, 1, 5, 1, 5, 5, 1, 10, 0, 5, 7, 2, 9, 2, 9, 0, 10, 7, 8, 0, 9, 10, 4, 9, 2, 7, 4, 2, 1, 4, 0, 10, 6, 0, 10, 5, 5, 2, 5, 6, 10, 0, 6, 6, 1, 10, 0, 3, 7, 5, 3, 10, 3, 10, 10, 5, 4, 8, 0, 5, 6, 4, 6, 8, 8, 0, 1, 8, 1, 10, 3
Offset: 1

Views

Author

Benoit Cloitre, Mar 09 2002

Keywords

Examples

			3.184809493b918664573b6211bb151551a05729...
		

Crossrefs

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), this sequence (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A060707 (b=60).
Cf. A007514.

Programs

  • Mathematica
    RealDigits[Pi,12,120][[1]] (* Harvey P. Dale, Jul 04 2014 *)
    Table[ResourceFunction["NthDigit"][Pi, n, 12], {n, 1, 120}] (* Joan Ludevid, Oct 11 2022; easy to compute a(10000000)=8 with this function; requires Mathematica 12.0+ *)

A096763 Position of the first occurrence of exactly n consecutive '9's in a row in the decimal expansion of Pi.

Original entry on oeis.org

5, 44, 2949, 17988, 19446, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 10542036048450, 5758910552709
Offset: 1

Views

Author

Robert G. Wilson v, Jul 07 2004

Keywords

Crossrefs

Cf. A000796: Decimal expansion (or digits) of Pi.
First occurrence of n times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
First occurrence of exactly n times the same digit: A096755 (exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.

Extensions

a(10)-a(11) from Giovanni Resta, Sep 30 2019
a(12) from Yasumasa Kanada, 2002 and a(13)-a(14) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019

A004605 Expansion of Pi in base 6.

Original entry on oeis.org

3, 0, 5, 0, 3, 3, 0, 0, 5, 1, 4, 1, 5, 1, 2, 4, 1, 0, 5, 2, 3, 4, 4, 1, 4, 0, 5, 3, 1, 2, 5, 3, 2, 1, 1, 0, 2, 3, 0, 1, 2, 1, 4, 4, 4, 2, 0, 0, 4, 1, 1, 5, 2, 5, 2, 5, 5, 3, 3, 1, 4, 2, 0, 3, 3, 3, 1, 3, 1, 1, 3, 5, 5, 3, 5, 1, 3, 1, 2, 3, 3, 4, 5, 5, 3, 3, 4, 1, 0, 0, 1, 5, 1, 5, 4, 3, 4, 4, 4, 0, 1, 2, 3, 4, 3
Offset: 1

Views

Author

Keywords

Examples

			3.05033005141512410523441405312532110230...
		

Crossrefs

Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), this sequence (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), A062964 (b=16), A060707 (b=60).
Cf. A007514.

Programs

  • Mathematica
    RealDigits[Pi, 6, 105][[1]]
    Table[ResourceFunction["NthDigit"][Pi, n, 6], {n, 1, 105}] (* Joan Ludevid ,Aug 17 2022;easy to compute a(10000000)=0 with this function;requires Mathematica 12.0+ *)

A019699 Decimal expansion of 2*Pi/15 = (4*Pi/3)/10.

Original entry on oeis.org

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

Views

Author

Keywords

Comments

With offset 1, decimal expansion of 4*Pi/3, the volume of a sphere of radius 1. - Omar E. Pol, Aug 27 2007, Sep 25 2013
2*Pi/15 is the common value of the base angles of the isosceles triangle formed at the common vertex of the figure obtained by gluing a hexagon and a pentagon, both regular, along a common side, as shown in the CNRS link. - Michel Marcus, Mar 06 2015
This is also the surface area (in some cubic length unit (l.u.)) of a sphere with a central cylinder symmetrical hole of length 2 l.u. Thanks to Sven Heinemeyer for reminding me of this classical astonishing result. See e.g., Bild der Wissenschaft, Januar 1964, p. 75, or the Gardner reference, Problem 7 on p. 51. In two dimensions things are different. See A258146. - Wolfdieter Lang, May 31 2015

Examples

			2*Pi/15 = 0.418879020478639098461685784437267...
4*Pi/3 = 4.18879020478639098461685784437267... - _Omar E. Pol_, Sep 25 2013
		

References

  • Bild der Wissenschaft, Januar 1964.
  • Martin Gardner, Mathematische Rätsel und Probleme, 3. Auflage, Friedr. Vieweg + Sohn, Braunschweig, 1975, p. 51 (in German). In English: Mathematical Puzzles and Diversions from "Scientific American", Simon and Schuster, N. Y. 1959/1961.

Crossrefs

Programs

Formula

(1/10)*volume of the unit sphere in R^3 = (1/10)*Pi^(3/2)/gamma(1+3/2). - Benoit Cloitre, Jun 19 2003
Previous Showing 81-90 of 1164 results. Next