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

%I A247548 #15 Dec 15 2024 07:20:45
%S A247548 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,
%T A247548 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,
%U A247548 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
%N A247548 Decimal expansion of D^2, a constant associated with the "Dimer Problem" on a triangular lattice.
%D A247548 Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.23 Monomer-dimer constants p. 408.
%F A247548 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).
%e A247548 2.35652735334624880922914314763999476796439150067841679838661876063419126231...
%t A247548 digits = 20; uv = Log[6 + 2*Cos[u] + 2*Cos[v] + 2*Cos[u + v]];
%t A247548 SetOptions[NIntegrate, WorkingPrecision -> digits + 5];
%t A247548 i1 = 2*NIntegrate[uv, {u, 0, Pi/2}, {v, 0, Pi/2}];
%t A247548 i2 = 4*NIntegrate[uv, {u, 0, Pi/2}, {v, Pi/2, Pi}];
%t A247548 i3 = 2*NIntegrate[uv, {u, -Pi, -Pi/2}, {v, Pi/2, Pi}];
%t A247548 i4 = 2*NIntegrate[uv, {u, -Pi/2, 0}, {v, 0, Pi/2}];
%t A247548 i5 = 4*NIntegrate[uv, {u, -Pi/2, 0}, {v, Pi/2, Pi}];
%t A247548 i6 = 2*NIntegrate[uv, {u, Pi/2, Pi}, {v, Pi/2, Pi}];
%t A247548 D2 = Exp[(1/(8*Pi^2))*(i1 + i2 + i3 + i4 + i5 + i6)];
%t A247548 RealDigits[D2, 10, digits] // First
%o A247548 (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
%Y A247548 Cf. A130834, A242710.
%K A247548 nonn,cons
%O A247548 1,1
%A A247548 _Jean-François Alcover_, Sep 19 2014
%E A247548 More terms from _Michel Marcus_, Sep 19 2014