A088538 -id:A088538 - cp's OEIS frontend

A079097 Mix odd numbers and squares.

1, 1, 3, 4, 5, 9, 7, 16, 9, 25, 11, 36, 13, 49, 15, 64, 17, 81, 19, 100, 21, 121, 23, 144, 25, 169, 27, 196, 29, 225, 31, 256, 33, 289, 35, 324, 37, 361, 39, 400, 41, 441, 43, 484, 45, 529, 47, 576, 49, 625, 51, 676, 53, 729, 55, 784, 57, 841, 59, 900, 61, 961, 63, 1024, 65

Offset: 0

N. J. A. Sloane, Sep 20 2008

The old entry with this sequence number was a duplicate of A010892.

This sequence is visible in the identity 4/Pi = 1 + 1/(3 + 4/(5 + 9/(7 + 16/(9 + 25/(11 + ...))))) (see the Wolfram Research web site).

Colin Barker, Table of n, a(n) for n = 0..1000
Wolfram Research, Pi
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A086377.

Cf. A088538 for decimal expansion of 4/Pi.

[(-1)*((1+n)*(-5-3*(-1)^n+(-1+(-1)^n)*n))/8: n in [0..70]]; // Vincenzo Librandi, Jan 28 2016

f:=n->if n mod 2 = 0 then n+1 else ((n+1)/2)^2; fi;

Riffle[2 Range@ Floor[#/2] - 1, Range[#]^2] &@66 (* or *) CoefficientList[Series[(1 + x^2) (x^2 - x - 1)/((x - 1)^3*(1 + x)^3), {x, 0, 64}], x] (* Michael De Vlieger, Jan 27 2016 *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 1, 3, 4, 5, 9}, 70] (* Vincenzo Librandi, Jan 28 2016 *)

Vec((1+x^2)*(x^2-x-1)/((x-1)^3*(1+x)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: (1+x^2)*(x^2-x-1)/((x-1)^3*(1+x)^3). - R. J. Mathar, Jan 05 2009
From Colin Barker, Jan 27 2016: (Start)
a(n) = (-1)*((1+n)*(-5-3*(-1)^n+(-1+(-1)^n)*n))/8.
a(n) = n-1 for n even.
a(n) = (n^2+2*n+1)/4 for n odd.
(End)

A132697 Decimal expansion of 7/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

			2.22816920328653470076437268721520106848243504036639028246734281682455516687917....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201.

Digits:=100; evalf(7/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[7/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

7/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132721 Decimal expansion of 31/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			9.867606471697510817670793...

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696, A132697, A132698, A132699, A132701, A132702.

RealDigits[31/Pi,10,120][[1]] (* Harvey P. Dale, Dec 28 2021 *)

31/Pi \\ Charles R Greathouse IV, Oct 01 2022

Terms a(32) and beyond from Andrew Howroyd, Jan 03 2020

A180310 Decimal expansion of Pi/2 - 4/Pi.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 26 2010

Torsional rigidity constant for a half-disk shaped shaft.

			0.29755678205973393308...

Index entries for transcendental numbers

Cf. A019669, A088538.

RealDigits[-4/Pi+Pi/2,10,120][[1]] (* Harvey P. Dale, May 02 2012 *)

Pi/2 - 4/Pi \\ Stefano Spezia, May 26 2025

Equals A019669 - A088538.

A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 08 2014

			0.822825249678847032995328716261464949475693118894850218393815613...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.10 Sierpinski's Constant, p. 123.

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants, and other related results, arXiv:1701.07064 (2017) Proposition 9.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 103.
Guillaume Melquiond, W. Georg Nowak, Paul Zimmermann, Numerical approximation of the Masser-Gramain constant to four decimal places, Mathematics of Computation, Volume 82, Number 282, April 2013, Pages 1235-1246
Eric Weisstein's MathWorld, Sierpiński's Constant
Wikipedia, Sierpiński's Constant

Cf. A062089, A062539, A068466, A086058.

S = Log[4*Pi^3*Exp[2*EulerGamma]/Gamma[1/4]^4]; RealDigits[S, 10, 104] // First

log(agm(sqrt(2), 1)^2/2) + 2*Euler \\ Charles R Greathouse IV, Nov 26 2024

S = gamma + beta'(1) / beta(1), where beta is Dirichlet's beta function.
S = log(Pi^2*exp(2*gamma) / (2*L^2)), where L is Gauss' lemniscate constant.
S = log(4*Pi^3*exp(2*gamma) / Gamma(1/4)^4), where gamma is Euler's constant and Gamma is Euler's Gamma function.
S = A062089 / Pi, where A062089 is Sierpiński's K constant.
S = A086058 - 1, where A086058 is the conjectured (but erroneous!) value of Masser-Gramain 'delta' constant. [updated by Vaclav Kotesovec, Apr 27 2015]
S = 2*gamma + (4/Pi)*integral_{x>0} exp(-x)*log(x)/(1-exp(-2*x)) dx.
Sum_{k=1..n} r(k)/k = Pi*(log(n) + S) + O(n^(-1/2)).
Equals 2*A001620 - A088538*A115252 [Coffey]. - R. J. Mathar, Jan 15 2021

A340533 Decimal expansion of log_2(4/Pi).

Original entry on oeis.org

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

Offset: 0

A.H.M. Smeets, Jan 10 2021

Probability of a coefficient in the continued fraction being even, where the continued fraction coefficients satisfy the Gauss-Kuzmin distribution.

			0.348503870527681201956720704891992664981523...

V. N. Nolte, Some probabilistic results on the convergents of continued fractions, Indagationes Mathematicae, Vol. 1, No. 3 (1990), pp. 381-389.

Cf. A088538, A216582, A340543.

RealDigits[Log[4/Pi]/Log[2], 10, 100][[1]] (* Amiram Eldar, Jan 10 2021 *)

PARI
```
log(4/Pi)/log(2)
```

Equals 2 - A216582.
Equals log_2(A088538).
Equals -Sum_{k >= 1} log_2(1-1/(2*k+1)^2).
Equals 1-A340543.

A132717 Decimal expansion of 27/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			8.59436692696234813151972322211577554986082086998464...

Index entries for transcendental numbers.

Cf. A049541, A060294, A089491, A088538, A086201, A132696, A132697, A132698, A132699, A132701, A132702.

RealDigits[27/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

27/Pi \\ Charles R Greathouse IV, Oct 01 2022

A240946 Decimal expansion of the average distance traveled in three steps of length 1 for a random walk in the plane starting at the origin.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 04 2014

			1.5745972375518936574946921830765...

J. M. Borwein, A. Straub, J. Wan, and W. Zudilin, Densities of short uniform random walks, arXiv:1103.2995 [math.CA], (11-August-2011)
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 464.

Cf. A088538 (two steps).

(3*2^(1/3))/(16*Pi^4)*Gamma[1/3]^6 + (27*2^(2/3))/(4*Pi^4)*Gamma[2/3]^6 //
  RealDigits[#, 10, 100]& // First (* updated May 20 2015 *)

Integral_(0..3) x*p(x) dx, where p(x) = 2*sqrt(3)/Pi*x/(3+x^2) * 2F1(1/3, 2/3; 1; x^2*(9-x^2)^2/(3+x^2)^3), 2F1 being the hypergeometric function.
Re(3F2(-1/2, -1/2, 1/2; 1, 1; 4)).
(3*2^(1/3))/(16*Pi^4)*Gamma(1/3)^6 + (27*2^(2/3))/(4*Pi^4)*Gamma(2/3)^6.

More digits from Jean-François Alcover, May 20 2015

A243378 Decimal expansion of a constant related to the asymptotic evaluation of Product_{p prime congruent to 3 modulo 4} (1 + 1/p).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 04 2014

			0.985247581006096343690510604298896801...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Ramanujan constant.

Cf. A001620, A064533, A088538, A088540.

digits = 102; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; 1/Sqrt[Pi]*Exp[EulerGamma/2]*1/LandauRamanujanK // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

Equals (1/sqrt(Pi))*exp(gamma/2)*1/K, where gamma is the Euler-Mascheroni constant (A001620) and K the Landau-Ramanujan constant (A064533).

Equals 4/(Pi*A088540) = A088538/A088540. - Amiram Eldar, Nov 16 2021

A331095 Decimal expansion of 32/Pi^3.

Original entry on oeis.org

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

Offset: 1

Dimitris Valianatos, Jan 08 2020

For odd prime numbers: Product_{odd primes p} 1/(1 - 1/p^2) = Pi^2/8 = (3/4)*zeta(2) = A111003.

For odd composite numbers: Product_{odd composite numbers c} 1/(1 - 1/c^2) = (81/80) * (225/224) * (441/440) * (625/624) * (729/728) * ... = 32/Pi^3, this constant.

			1.032049101862383653901505686034038...

Index entries for transcendental numbers

Cf. A088538 (4/Pi), A111003 (Pi^2/8).

RealDigits[32/Pi^3, 10, 100][[1]] (* Amiram Eldar, Jan 10 2020 *)

p = 1.0; forstep(n = 3, 10^7, 2, if(!isprime(n), p*= (1 / (1 - 1 / n^2)))); print(p)

PARI
```
32/Pi^3
```

Equals A088538/A111003.

cp's OEIS Frontend

A079097 Mix odd numbers and squares.

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A132697 Decimal expansion of 7/Pi.

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A132721 Decimal expansion of 31/Pi.

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A180310 Decimal expansion of Pi/2 - 4/Pi.

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A241017 Decimal expansion of Sierpiński's S constant, which appears in a series involving the function r(n), defined as the number of representations of the positive integer n as a sum of two squares. This S constant is the usual Sierpiński K constant divided by Pi.

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A340533 Decimal expansion of log_2(4/Pi).

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A132717 Decimal expansion of 27/Pi.

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