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

Author

Jean-François Alcover, Sep 19 2014

Keywords

• nonn
• cons

Examples

			2.35652735334624880922914314763999476796439150067841679838661876063419126231...
		

References

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

Crossrefs

Cf. A130834, A242710.

Programs

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

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

Formula

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

Extensions

More terms from Michel Marcus, Sep 19 2014