A143233 -id:A143233 - cp's OEIS frontend

A097469 Decimal expansion of growth constant C for dimer model on square grid.

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

Offset: 1

Ralf Stephan, Sep 18 2004

			1.33851515197609676693819590201851353706435369712791131464123...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.8.3 and 5.23.1, pp. 63, 407.

G. C. Greubel, Table of n, a(n) for n = 1..10000
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
J. Propp, Enumeration of matchings: problems and progress, in New Perspectives in Algebraic Combinatorics

Cf. A038758, A054344, A130834, A143233.

Cf. A000796 (Pi), A006752 (Catalan's constant).

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

RealDigits[Exp[Catalan/Pi], 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

default(realprecision, 100); exp(Catalan/Pi) \\ G. C. Greubel, Aug 25 2018

Equals e^(G/Pi), with G = A006752 (Catalan's constant).
Equals exp((1/Pi^2) * Integral_{x=0..Pi/2, y=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2) dx dy). - Vaclav Kotesovec, Jan 04 2021
Equals sqrt(A130834) = exp(A143233). - Hugo Pfoertner, Nov 18 2024

Terms a(14) onward corrected by G. C. Greubel, Aug 26 2018

A256318 Decimal expansion of Sum_{k>=0} zeta(2k)/((2k+1)*4^(2k)) (negated).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 23 2015

			-0.464847699170805107492692486832939060882941187575901...

G. C. Greubel, Table of n, a(n) for n = 0..10000
H. M. Srivasata, M. L. Glasser, and Victor S. Adamchik, Some Definite Integrals Associated with the Riemann Zeta Function

Cf. A006752, A256319.

SetDefaultRealField(RealField(100)); R:=RealField(); Catalan(R)/Pi(R) + Log(2)/4; // G. C. Greubel, Aug 25 2018

RealDigits[-Catalan/Pi - Log[2]/4, 10, 105] // First

Catalan/Pi + log(2)/4 \\ Charles R Greathouse IV, Mar 23 2015

 5 - sumpos(k=1,zeta(2*k)/(2*k+1)/16^k) \\ Charles R Greathouse IV, Mar 23 2015

Equals -G/Pi - log(2)/4 = -A143233-A379101, where G is Catalan's constant.

A345738 Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 25 2021

A projectile is launched with an initial speed v at angle theta above the horizon. Assuming that the gravitational acceleration g is uniform and neglecting the air resistance, the trajectory is a part of a parabola whose expected length, averaged over theta uniformly chosen at random from the range [0, Pi/2], is c * v^2/g, where c is this constant.

The length of the trajectory as a function of theta is L(theta) = (v^2/g)*(sin(theta) + cos(theta)^2*log((1+sin(theta))/(1-sin(theta)))/2). L(theta) goes from 0 to 1 between theta = 0 and Pi/2. It has a maximum at theta = 0.985514... (A345737), and a unique value at 0 <= theta < 0.599677... (A345739). The average length (c * v^2/g) occurs at theta = 0.5152731296... (29.522975... degrees).

			0.90143169424542823181453643968181856179705159945258...

Péter Kórus, Notes on Projectile Motion, The American Mathematical Monthly, Vol. 126, No. 4 (2019), pp. 358-360.

Cf. A000796, A006752, A143233, A345737, A345739.

RealDigits[(2*Catalan + 1)/Pi, 10, 100][[1]]

Equals (2 * A006752 + 1)/A000796.

Equals 2 * A143233 + 1.

A214174 Decimal expansion of Integral_{x=0..oo} x/cosh(Pi*x) dx.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 20 2013

This sequence is between A020761 [Integral_{x=0..oo} x^0/cosh(Pi*x) dx] and A020821 [Integral_{x=0..oo} x^2/cosh(Pi*x) dx].

			0.1856134363553860814769819901997199856094304591312932054076019...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A006752, A020761, A020821, A060294, A143233.

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

Mathematica
```
RealDigits[2 Catalan/Pi^2, 10, 90][[1]]
```

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

Equals A060294*A143233 = 2*C/Pi^2, where C is Catalan's constant (A006752).

A322757 Decimal expansion of G/(2*Pi), where G is Catalan's constant A006752.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 28 2018

Per-site entropy of the dimer model on a square grid.

			0.1457804520154093900691922282341974594320330769929186351305007845558738184...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 554.

Cf. A006752, A019692, A143233.

RealDigits[Catalan/(2*Pi), 10, 120][[1]] (* Amiram Eldar, May 24 2023 *)

Equals (1/2) * A143233. - Peter Bala, Aug 01 2025

A378910 Decimal expansion of 2*G/Pi, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 10 2024

			0.58312180806163756027676891293678983772813230797167...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 232.

Cf. A000796, A006752, A060294, A130834, A143233, A221209, A378911.

Mathematica
```
RealDigits[2Catalan/Pi,10,100][[1]]
```

Equals log(A130834).

Equals 2*A143233.

A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 10 2024

			2.5337372794858419095832896340418632916896308088420...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 233.

Cf. A000796, A002193, A006752, A060294, A130834, A143233, A221209, A378910.

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

A348371 Decimal expansion of Sum_{k>=0} binomial(2k,k)^2/(16^k(k+1)^3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 15 2021

			1.03928049679487622006025262010356644086601121330111...

Pablo Fernandez Refolio, Problem 12094, The American Mathematical Monthly, Vol. 126, No. 2 (2019), p. 180; Catalan's Constant from Catalan Numbers, Solution to Problem 12094 by Roberto Tauraso, ibid., Vol. 127, No. 8 (2020), pp. 755-757.

Cf. A000984, A006752, A143233, A244009.

RealDigits[48/Pi - 16*(1 - Log[2]) - 32*Catalan/Pi, 10, 100][[1]]

Equals 48/Pi - 16*(1 - log(2)) - 32*G/Pi, where G is Catalan's constant (A006752).

Equals 4F3(1/2, 1/2, 1, 1; 2, 2, 2; 1), where pFq() is the generalized hypergeometric function.

A378025 Decimal expansion of 1/2 - log(2)/4 - G/Pi, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 14 2024

			0.035152300829194892507307513167060939117058812424...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 56.

Cf. A000796, A002162, A006752, A143233.

RealDigits[1/2-Log[2]/4-Catalan/Pi,10,100,-1][[1]]

Equals Sum_{n>=1} zeta(2*n)/(2^(4*n)*(2*n + 1)) (see Finch).

cp's OEIS Frontend

A097469 Decimal expansion of growth constant C for dimer model on square grid.

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A256318 Decimal expansion of Sum_{k>=0} zeta(2k)/((2k+1)*4^(2k)) (negated).

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A345738 Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

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A214174 Decimal expansion of Integral_{x=0..oo} x/cosh(Pi*x) dx.

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A322757 Decimal expansion of G/(2*Pi), where G is Catalan's constant A006752.

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A378910 Decimal expansion of 2*G/Pi, where G = A006752.

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A378911 Decimal expansion of sqrt(2)*exp(2*G/Pi), where G = A006752.

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A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

A348371 Decimal expansion of Sum_{k>=0} binomial(2k,k)^2/(16^k(k+1)^3).