A242967 -id:A242967 - cp's OEIS frontend

A245725 Decimal expansion of z_tri, a constant related to the enumeration of spanning trees on the triangular lattice (this is different from A242968).

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

Offset: 1

Jean-François Alcover, Jul 30 2014

			1.6153297360972525704681825536190319703612092039029350806543423518...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.22 Lenz-Ising Constants, p. 400.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Robert Shrock and F. Y. Wu, Spanning Trees on Graphs and Lattices in d Dimensions, arXiv:cond-mat/0004341 [cond-mat.stat-mech], 2000, p. 7.

Cf. A242967, A242968, A242969.

H = Sqrt[3]/(6*Pi)*PolyGamma[1, 1/6] - Pi/Sqrt[3] - Log[6]; RealDigits[Log[2] + Log[3] + H, 10, 104] // First
(* or *) 3*(Sqrt[3]/Pi)*N[Sum[1/n^2 - 1/(n+4)^2, {n, 1, Infinity, 6}], 104] // RealDigits // First

Equals log(2) + log(3) + H, where H is the auxiliary constant A242967.

Equals Sum_{n>=1} 10*n*(arccoth((3*n) / 2) - 2 * arccoth(3*n)). - Antonio Graciá Llorente, Oct 13 2024

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.

A242968 Decimal expansion of an Ising constant related to the triangular lattice.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 28 2014

			0.879585386161571517093896028307972843...

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

Cf. A242967, A242969.

H = Sqrt[3]/(6*Pi)*PolyGamma[1, 1/6]  -  Pi/Sqrt[3] - Log[6]; Log[2] + Log[3]/4 + H/2 // RealDigits[#, 10, 103] & // First

log(2) + log(3)/4 + H/2, where H is defined in A242967.

A242969 Decimal expansion of an Ising constant related to the hexagonal lattice.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 28 2014

			1.0250590964713132201744730083644777...

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

Cf. A242967, A242968.

H = Sqrt[3]/(6*Pi)*PolyGamma[1, 1/6] - Pi/Sqrt[3] - Log[6]; 3/4*Log[2] + Log[3]/2 + H/4 // RealDigits[#, 10, 100]& // First

3/4*log(2) + log(3)/2 + H/4, where H is defined in A242967.

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)).

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)).

cp's OEIS Frontend

A245725 Decimal expansion of z_tri, a constant related to the enumeration of spanning trees on the triangular lattice (this is different from A242968).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A242968 Decimal expansion of an Ising constant related to the triangular lattice.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

A242969 Decimal expansion of an Ising constant related to the hexagonal lattice.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula