A218387 -id:A218387 - cp's OEIS frontend

A245736 Decimal expansion of z_br = z_4.8.8, the bulk limit of the number of spanning trees on a "bathroom" lattice (squares and octagons).

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

Offset: 0

Jean-François Alcover, Jul 31 2014

			0.786684275378832179121657989494695380551170816578...

Shu-Chiuan Chang and Robert Shrock, Some Exact Results for Spanning Trees on Lattices., Discrete Math., J. Phys. A: Math. Gen. 39, 5653-5658 (2006).

Cf. A218387(z_sq), A245725(z_tri).

z[br] = Catalan/Pi + (1/4)*Log[3-2*Sqrt[2]] + (1/Pi)*Integrate[ArcTan[t]/t, {t, 0, 3+2*Sqrt[2]}]; RealDigits[N[Re[z[br]], 103]] // First

C/Pi + (1/4)*log(3-2*sqrt(2)) + (1/Pi)*integral_{0..3+2*sqrt(2)} arctan(t)/t dt, where C is Catalan's constant.

A245737 Decimal expansion of z_hc, the bulk limit of the number of spanning trees on a honeycomb lattice.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 31 2014

			0.8076648680486262852340912768095159851806046019514675403271711759...

Robert Shrock and F. Y. Wu, Spanning Trees on Graphs and Lattices in d Dimensions p. 7.

Cf. A218387(z_sq), A242967(H), A245725(z_tri).

H = Sqrt[3]/(6*Pi)*PolyGamma[1, 1/6] - Pi/Sqrt[3] - Log[6]; RealDigits[(1/2)*(Log[2] + Log[3] + H), 10, 103] // First

(1/2)*(log(2) + log(3) + H), where H is the auxiliary constant A242967.

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

Original entry on oeis.org

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.

A245739 Decimal expansion of z_kag, the bulk limit of the number of spanning trees on a kagomé lattice.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 31 2014

			1.1356964017751025237602199706665780810280666320286465955...

Robert Shrock and F. Y. Wu, Spanning Trees on Graphs and Lattices in d Dimensions pp. 21-25.
Wikipedia, Trihexagonal tiling [Kagomé lattice]

Cf. A218387(z_sq), A242967(H), A245725(z_tri), A245736(z_br), A245737(z_hc).

H = Sqrt[3]/(6*Pi)*PolyGamma[1, 1/6] - Pi/Sqrt[3] - Log[6]; RealDigits[(1/3)*(2*Log[2] + 2*Log[3] + H), 10, 103] // First

(1/3)*(2*log(2) + 2*log(3) + H), where H is the auxiliary constant A242967.

Equals (1/3)*(A245725 + log(6)).

A377753 Decimal expansion of 8*G/Pi^2, where G is the Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 07 2024

			0.7424537454215443259079279607988799424377218365...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 7.7, pp. 54, 474.

Cf. A002388, A006752, A218387, A242822 (see the second formula).

Mathematica
```
RealDigits[8Catalan/Pi^2,10,100][[1]]
```

8*Catalan/Pi^2 \\ Charles R Greathouse IV, Nov 27 2024

Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} 1/(2*n - 1)^2) (see Finch).
Equals 1/A242822. - Hugo Pfoertner, Nov 07 2024
Equals (1 - W)/(1 + W), where W = tanh(Sum_{prime p == 3 (mod 4)} arctanh(1/p^2)) = zeta(2,3/4)/zeta(2,1/4) = (Pi^2 - 8*G)/(Pi^2 + 8*G) = 0.1478066521164... Physical interpretation: the constant W is the relativistic sum of the velocities c/p^2 over all primes p == 3 (mod 4), in units where the speed of light c = 1. - Thomas Ordowski, Nov 23 2024

A245724 Decimal expansion of 'g', a constant related to the asymptotic distribution of the Q moment and the logarithmic divergence of the Ising specific heat.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 30 2014

			0.619403698458631885718514208104164609712358495509146535066838397776...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Ising Model

Cf. A218387, A242743.

g = 1 + Pi/4 + Log[(Sqrt[2]/4)*Log[1 + Sqrt[2]]]; RealDigits[g, 10, 102] // First

g = 1 + Pi/4 + log((sqrt(2)/4)*log(1 + sqrt(2))).

A245740 Decimal expansion of z_(3-12-12), the bulk limit of the number of spanning trees on a 3-12-12 lattice.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 31 2014

			0.720563322866577106077364520627957552422383519332367042383614...

Robert Shrock and F. Y. Wu, Spanning Trees on Graphs and Lattices in d Dimensions pp. 21-25.
Wikipedia, Truncated hexagonal tiling

Cf. A218387(z_sq), A242967(H), A245725(z_tri), A245736(z_br), A245737(z_hc), A245739(z_kag).

H = Sqrt[3]/(6*Pi)*PolyGamma[1, 1/6] - Pi/Sqrt[3] - Log[6]; RealDigits[(1/6)*(Log[2] + 2*Log[3] + Log[5] + H), 10, 103] // First

(1/6)*(log(2) + 2*log(3) + log(5) + H), where H is the auxiliary constant A242967.

Equals (1/6)*(A245725 + log(15)).

A245741 Decimal expansion of z_UJ, the bulk limit of the number of spanning trees on a Union Jack lattice.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 31 2014

			1.5733685507576643582433159789893907611023416331560654994372929...

Robert Shrock and F. Y. Wu, Spanning Trees on Graphs and Lattices in d Dimensions pp. 21-25.

Cf. A218387(z_sq), A245725(z_tri), A245736(z_br), A245737(z_hc), A245739(z_kag), A245740(z_(3-12-12)).

z[UJ] = 2*Catalan/Pi + (1/2)*Log[3 - 2*Sqrt[2]] + (2/Pi)*Integrate[ArcTan[t]/t, {t, 0, 3 + 2*Sqrt[2]}]; RealDigits[N[Re[z[UJ]], 104]] // First

2*C/Pi + (1/2)*log(3-2*sqrt(2)) + (2/Pi)*integral_{0..3+2*sqrt(2)} arctan(t)/t dt, where C is Catalan's constant.

Equals 2*A245736.

A370374 Decimal expansion of 2log(2) - 4Catalan/Pi.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jun 07 2024

			0.22005074499...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556, L*(0).

Cf. A016627, A218387.

Maple
```
2*log(2)-4*Catalan/Pi ; evalf(%) ;
```

RealDigits[2*Log[2] - 4*Catalan/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

2*log(2) - 4*Catalan/Pi \\ Amiram Eldar, Jun 10 2024

Equals Sum_{n>=1} ((2n-1)!!/(2n)!!)^2 / (2*n).
Equals A016627 - A218387.
Equals Sum_{k>=1} binomial(2*k,k)^2/(k*2^(4*k+1)) (see Finch). - Stefano Spezia, Nov 13 2024

cp's OEIS Frontend

A245736 Decimal expansion of z_br = z_4.8.8, the bulk limit of the number of spanning trees on a "bathroom" lattice (squares and octagons).

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A245737 Decimal expansion of z_hc, the bulk limit of the number of spanning trees on a honeycomb lattice.

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

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A245739 Decimal expansion of z_kag, the bulk limit of the number of spanning trees on a kagomé lattice.

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A377753 Decimal expansion of 8*G/Pi^2, where G is the Catalan's constant (A006752).

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A245724 Decimal expansion of 'g', a constant related to the asymptotic distribution of the Q moment and the logarithmic divergence of the Ising specific heat.

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A245740 Decimal expansion of z_(3-12-12), the bulk limit of the number of spanning trees on a 3-12-12 lattice.

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A245741 Decimal expansion of z_UJ, the bulk limit of the number of spanning trees on a Union Jack lattice.

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A370374 Decimal expansion of 2log(2) - 4Catalan/Pi.