cp's OEIS Frontend

A355954 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.

%I A355954 #24 Aug 14 2022 15:29:28
%S A355954 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 A355954 8,8,1,5,4,5,6,5,1
%N A355954 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 A355954 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 A355954 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 A355954 0.3344120313924198...
%t A355954 alphat[beta_] := ArcCosh[2/Cos[beta] - Cos[beta]];
%t A355954 Rtri[n_, p_] :=
%t A355954   SetAccuracy[1/(Pi), 150]*
%t A355954    NIntegrate[(1 -
%t A355954        Exp[-Abs[n - p]*alphat[beta]]*Cos[(n + p)*beta])/(Cos[
%t A355954         beta]*Sinh[alphat[beta]]), {beta, 0, Pi/2},
%t A355954     WorkingPrecision -> 150];
%t A355954 Rtri[3*10^8, 0] - SetAccuracy[Log[3*10^8]/(Pi* Sqrt[3]), 150];
%Y A355954 Cf. A355589, A355953 (similar for square lattice).
%Y A355954 Cf. A355585, A355586, A355587, A355588 (exact solutions for small distances).
%K A355954 nonn,cons,hard,more
%O A355954 0,1
%A A355954 _Hugo Pfoertner_, Jul 26 2022