OFFSET
1,1
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.23 Monomer-dimer constants p. 408.
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
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Sep 19 2014
EXTENSIONS
More terms from Michel Marcus, Sep 19 2014
STATUS
approved