A245592 -id:A245592 - cp's OEIS frontend

A242743 Decimal expansion of an Ising constant related to the random coloring problem.

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

Offset: 0

Jean-François Alcover, Jun 02 2014

In Ising model on the 2D square lattice, the negated ratio of free energy per node to the temperature at the critical point. - Andrey Zabolotskiy, Sep 12 2017

			0.929695398341610214985384973665878...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.22 Lenz-Ising constants, p. 399.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Lars Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944) [see eq. (121)].

Cf. A002898, A006752, A218387, A245592.

SetDefaultRealField(RealField(100)); R:=RealField(); Log(2)/2 + 2*Catalan(R)/Pi(R); // G. C. Greubel, Aug 25 2018

RealDigits[Log[2]/2 + 2*Catalan/Pi, 10, 105] // First

default(realprecision, 100); log(2)/2 + 2*Catalan/Pi \\ G. C. Greubel, Aug 25 2018

log(2)/2 + 2*G/Pi = log(2)/2 + A218387/2, where G is Catalan's constant.

A278928 Decimal expansion of sqrt(sqrt(2) + 1).

Original entry on oeis.org

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

Offset: 1

Bobby Jacobs, Dec 01 2016

A quartic integer with minimal polynomial x^4 - 2*x^2 - 1. - Charles R Greathouse IV, Dec 01 2016
Suppose f(n) has the recurrence f(2*n) = f(2*n - 1) + f(2*n - 2) and f(2*n + 1) = f(2*n) + f(2*n - 2), where f(0) and f(1) are not both 0. Then, lim_{n -> oo} f(n)^(1/n) is this constant.
Apart from the first digit, the same as A190283. - R. J. Mathar, Dec 09 2016
Imaginary part of sqrt(1 + i)^3, where i is the imaginary unit such that i^2 = -1. See A154747 for real part. - Alonso del Arte, Sep 09 2019

			1.553773974030037307344158953063146948164583499410307836332671...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 7.4, p. 466.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4.

Cf. A002965, A014176, A154747, A190283, A245592.

Cf. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).

Sqrt(1+Sqrt(2)); // G. C. Greubel, Apr 14 2018

Digits:=100: evalf(sqrt(sqrt(2)+1)); # Wesley Ivan Hurt, Dec 01 2016

RealDigits[Sqrt[Sqrt[2] + 1], 10, 100][[1]] (* Wesley Ivan Hurt, Dec 01 2016 *)

sqrt(sqrt(2)+1) \\ Charles R Greathouse IV, Dec 01 2016

polrootsreal(x^4 - 2*x^2 - 1)[2] \\ Charles R Greathouse IV, Dec 01 2016

Equals 1/A154747.
Limit_{n -> oo} A002965(n)^(1/n).
From Peter Bala, Jul 01 2024: (Start)
This constant occurs in the evaluation of Integral_{x = 0..Pi/2} 1/(1 + sin^4(x)) dx = Pi/4 * sqrt(sqrt(2) + 1).
Equals 2*Sum_{n >= 0} (-1/16)^n * binomial(4*n, 2*n) (a slowly converging series). (End)
Equals 2^(3/4)*cos(Pi/8). - Vaclav Kotesovec, Jul 01 2024
Equals Product_{k>=0} coth(Pi/4 + k*Pi/2). - Antonio Graciá Llorente, Dec 19 2024
Equals sqrt(A014176) = 1/A154747 = exp(A245592). - Hugo Pfoertner, Dec 19 2024

cp's OEIS Frontend

A242743 Decimal expansion of an Ising constant related to the random coloring problem.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A278928 Decimal expansion of sqrt(sqrt(2) + 1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula