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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

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: A191741 A191665 A254105 * A001600 A175578 A000036

Adjacent sequences:  A247545 A247546 A247547 * A247549 A247550 A247551

KEYWORD

nonn,cons

AUTHOR

Jean-Fran├žois Alcover, Sep 19 2014

EXTENSIONS

More terms from Michel Marcus, Sep 19 2014

STATUS

approved

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Last modified February 23 23:29 EST 2018. Contains 299595 sequences. (Running on oeis4.)