A355954 - OEIS

%I #24 Aug 14 2022 15:29:28

%S 3,3,4,4,1,2,0,3,1,3,9,2,4,1,9,8,0,2,0,4,3,9,1,3,9,1,2,2,2,1,2,2,7,0,

%T 8,8,1,5,4,5,6,5,1

%N Decimal expansion of the constant A in the asymptotic behavior R(d) = log(d)/(Pi*sqrt(3)) + A of the resistance between two nodes separated by the Euclidean distance d in an infinite triangular lattice of one-ohm resistors.

%C From an engineering point of view, this constant summand can be regarded as a kind of near-field contribution, which contains the well-known resistance of 1/3 ohms between 2 neighboring nodes as the main part.

%C The asymptotic formula is analogous to that known for the square lattice. The constant was determined by comparison with the exact integral (see A355589) for the resistance, evaluated for very large distances d (maximum approx. 10^9, for larger arguments the computational effort is no longer manageable). At the moment (July 2022) no representation in closed form is known. A derivation similar to the method used to determine A355953 might be applicable.

%e 0.3344120313924198...

%t alphat[beta_] := ArcCosh[2/Cos[beta] - Cos[beta]];

%t Rtri[n_, p_] :=

%t   SetAccuracy[1/(Pi), 150]*

%t    NIntegrate[(1 -

%t        Exp[-Abs[n - p]*alphat[beta]]*Cos[(n + p)*beta])/(Cos[

%t         beta]*Sinh[alphat[beta]]), {beta, 0, Pi/2},

%t     WorkingPrecision -> 150];

%t Rtri[3*10^8, 0] - SetAccuracy[Log[3*10^8]/(Pi* Sqrt[3]), 150];

%Y Cf. A355589, A355953 (similar for square lattice).

%Y Cf. A355585, A355586, A355587, A355588 (exact solutions for small distances).

%K nonn,cons,hard,more

%O 0,1

%A _Hugo Pfoertner_, Jul 26 2022