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 A247548 Decimal expansion of D^2, a constant associated with the "Dimer Problem" on a triangular lattice. 0
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 REFERENCES Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.23 Monomer-dimer constants p. 408. LINKS 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 A316609 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

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Last modified January 21 03:58 EST 2022. Contains 350473 sequences. (Running on oeis4.)