A049006 -id:A049006 - cp's OEIS frontend

A330865 Decimal expansion of cosh(Pi/2)/Pi.

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			(1 - 1/2^2) * (1 + 1/3^2) * (1 - 1/4^2) * (1 + 1/5^2) * (1 - 1/6^2) * ... = (e^(Pi/2) + e^(-Pi/2))/(2*Pi) = 0.7986963159564630848638067...

Cf. A042972, A049006, A049541, A156648, A175615, A308715, A308716, A308718, A323982, A330864, A334402.

RealDigits[Cosh[Pi/2]/Pi, 10, 110] [[1]]

cosh(Pi/2)/Pi \\ Michel Marcus, Apr 28 2020

Equals Sum_{k>=0} Pi^(2*k-1)/(4^k*(2*k)!).
Equals Product_{k>=2} (1 - (-1)^k/k^2).
Equals (i^(-i) + i^i)/(2*Pi), where i is the imaginary unit.

A383851 Decimal expansion of exp(8G/Pi)((1 - exp(-Pi/2))/(1 + exp(-Pi/2)))^2, where G is Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 13 2025

			4.4312013071941991970823677283552872932838015281012...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Michael Penn, A beautiful Ramanujan product, YouTube video, 2025.

Cf. A006752, A049006, A377753.

First[RealDigits[Exp[8*Catalan/Pi]*((1 - #)/(1 + #))^2 & [Exp[-Pi/2]], 10, 100]]

Equals Product_{i=0..oo} (1 + 4/(2*i+1)^4)^((-1)^i*(2*i+1)) (from Ramanujan).

A083283 Engel expansion for i^i = exp(-Pi/2).

Original entry on oeis.org

5, 26, 42, 45, 84, 2577, 6837, 18441, 43190, 65404, 502536, 2987594, 5828519, 41429808, 431610158, 1524989277, 4301621811, 5681026585, 21585646103, 26950130930, 174893227975, 334043333925, 453875893331, 633636954317, 17921965531082, 29347098990308

Offset: 1

Hauke Worpel (hw1(AT)email.com), Jun 03 2003

nonn

Cf. A049006, A049007.

Offset and name corrected by Alois P. Heinz, Nov 22 2020

A321161 Decimal expansion of Wilf's formula: Product_{k>=1} exp(-1/k)(1 + 1/k + 1/(2k^2)) = exp(-gamma)*cosh(Pi/2)/(Pi/2).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jan 11 2019

The formula was discovered by Wilf in 1997.

			0.896871242167379021690230319086367005622530649081704...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2011, p. 366.

Chao-Ping Chen and Richard B. Paris, Generalizations of two infinite product formulas, Integral Transforms and Special Functions, Vol. 25, No. 7 (2014), pp. 547-551.
Chao-Ping Chen and Richard B. Paris, On the asymptotics of products related to generalizations of the Wilf and Mortini problems, Integral Transforms and Special Functions, Vol. 27, No. 4 (2016), pp. 281-288, alternative link.
Chao-Ping Chen and Richard B. Paris, On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas, Applied Mathematics and Computation, Vol. 293 (2017), pp. 30-39, preprint, arXiv:1511.09217 [math.CA], 2015.
Junesang Choi, Jungseob Lee and H.M. Srivastava, A generalization of Wilf's formula, Kodai Mathematical Journal, Vol. 26, No. 1 (2003), pp. 44-48.
Junesang Choi and H.M. Srivastava, Integral Representations for the Euler-Mascheroni Constant gamma, Integral Transforms and Special Functions, Vol. 21, No. 9 (2010), pp. 675-690.
J. Lopez-Bonilla and R. López-Vázquez, On an Identity of Wilf for the Euler-Mascheroni's Constant, Prespacetime Journal, Vol. 9, No. 6 (2018), pp. 516-518.
Herbert S. Wilf, Problem 10588, The American Mathematical Monthly, Vol. 104, No. 5 (1997), p. 456.

Cf. A001620, A042972, A049006, A073004, A080130.

RealDigits[Exp[-EulerGamma]*Cosh[Pi/2]/(Pi/2), 10, 100][[1]]

exp(-Euler)*cosh(Pi/2)/(Pi/2) \\ Michel Marcus, Jan 15 2019

A334624 Decimal expansion of Pi + e + phi + sqrt(2) + i^i - 1/10.

Original entry on oeis.org

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

Offset: 1

Michal Paulovic, Sep 09 2020

A near-integer obtained by subtracting one tenth from the sum of Archimedes's constant, Napier's constant, golden ratio, Pythagoras's constant and the imaginary unit to the power of itself.

			9.0000016095...

Cf. A000796, A001113, A001622, A002193, A049006, A133055.

pi + exp(1) + (sqrt(5)+1)/2 + sqrt(2) + i^i - 1/10

Digits:=100; evalf(Pi + exp(1) + (sqrt(5)+1)/2 + sqrt(2) + I^I - 1/10);

RealDigits[Pi + Exp[1] + GoldenRatio + Sqrt[2] + Re[I^I] - 1/10, 10, 100][[1]]

Pi + exp(1) + (sqrt(5)+1)/2 + sqrt(2) + real(I^I) - 1/10

Equals Pi + e + (sqrt(5)+1)/2 + sqrt(2) + e^(-Pi/2) - 1/10.

A356983 Decimal expansion of Pi * e^(-Pi/2).

Original entry on oeis.org

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

Offset: 0

Christoph B. Kassir, Sep 07 2022

			0.653072949894912131389531881117225431...

Michael Penn, This integral looks crazy, YouTube video, 2022.

Cf. A000796, A001113, A049006.

RealDigits[Pi * Exp[-Pi/2], 10, 100][[1]]

PARI
```
Pi * exp(-Pi/2)
```

Equals Integral_{x=0..Pi} i^tan(x) dx, where i is the imaginary unit.

cp's OEIS Frontend

A330865 Decimal expansion of cosh(Pi/2)/Pi.

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A383851 Decimal expansion of exp(8*G/Pi)*((1 - exp(-Pi/2))/(1 + exp(-Pi/2)))^2, where G is Catalan's constant (A006752).

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A083283 Engel expansion for i^i = exp(-Pi/2).

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A321161 Decimal expansion of Wilf's formula: Product_{k>=1} exp(-1/k)*(1 + 1/k + 1/(2*k^2)) = exp(-gamma)*cosh(Pi/2)/(Pi/2).

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A334624 Decimal expansion of Pi + e + phi + sqrt(2) + i^i - 1/10.

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A356983 Decimal expansion of Pi * e^(-Pi/2).

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A383851 Decimal expansion of exp(8G/Pi)((1 - exp(-Pi/2))/(1 + exp(-Pi/2)))^2, where G is Catalan's constant (A006752).

A321161 Decimal expansion of Wilf's formula: Product_{k>=1} exp(-1/k)(1 + 1/k + 1/(2k^2)) = exp(-gamma)*cosh(Pi/2)/(Pi/2).