A247548 Decimal expansion of D^2, a constant associated with the "Dimer Problem" on a triangular lattice.

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

Offset

1,1

References

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.23 Monomer-dimer constants p. 408.

Links

Table of n, a(n) for n=1..104.

Formula

exp( 1/(8*Pi^2) * integral_{-Pi..Pi} integral_{-Pi..Pi} log(6 + 2*cos(u) + 2*cos(v) + 2*cos(u+v)) du dv).

Example

2.35652735334624880922914314763999476796439150067841679838661876063419126231...

Mathematica

digits = 20; uv = Log[6 + 2*Cos[u] + 2*Cos[v] + 2*Cos[u + v]];
SetOptions[NIntegrate, WorkingPrecision -> digits + 5];
i1 = 2*NIntegrate[uv, {u, 0, Pi/2}, {v, 0, Pi/2}];
i2 = 4*NIntegrate[uv, {u, 0, Pi/2}, {v, Pi/2, Pi}];
i3 = 2*NIntegrate[uv, {u, -Pi, -Pi/2}, {v, Pi/2, Pi}];
i4 = 2*NIntegrate[uv, {u, -Pi/2, 0}, {v, 0, Pi/2}];
i5 = 4*NIntegrate[uv, {u, -Pi/2, 0}, {v, Pi/2, Pi}];
i6 = 2*NIntegrate[uv, {u, Pi/2, Pi}, {v, Pi/2, Pi}];
D2 = Exp[(1/(8*Pi^2))*(i1 + i2 + i3 + i4 + i5 + i6)];
RealDigits[D2, 10, digits] // First

Prog

(PARI) exp(1/(8*Pi^2) * intnum(u=-Pi, Pi, intnum(v=-Pi, Pi, log(6 + 2*cos(u) + 2*cos(v) + 2*cos(u+v))))) \\ Michel Marcus, Sep 19 2014

Crossrefs

Cf. A130834, A242710.

Sequence in context: A191665 A306233 A254105 * A001600 A175578 A347861

Adjacent sequences: A247545 A247546 A247547 * A247549 A247550 A247551

Keyword

nonn,cons

Author

Jean-François Alcover, Sep 19 2014

Extensions

More terms from Michel Marcus, Sep 19 2014

Status

approved