A355955 - OEIS

%I #8 Jul 24 2022 10:43:24

%S 1,5,107,2460,56922,1317211,30481165,705355254,16322409116

%N a(n) is the least distance of two nodes on the same grid line in an infinite square lattice of one-ohm resistors for which the resistance measured between the two nodes is greater than n ohms.

%C The terms are obtained by a high-precision evaluation of the integral R(j,k) = (1/Pi) * Integral_{beta=0..Pi} (1 - exp(-abs(j)*alphas(beta))*cos(k*beta)) / sinh(alphas(beta)), with alphas(beta) = log(2 - cos(beta) + sqrt(3 + cos(beta)*(cos(beta) - 4))) such that floor(R(m-1,0)) < floor(R(m,0)). The values of m for which this condition is satisfied are the terms of the sequence. See Atkinson and van Steenwijk (1999, page 491, Appendix B) for a Mathematica implementation of the integral.

%C a(9) = 377711852375, found by solving R(x) - 9 = 0, using the asymptotic formula provided by Cserti (2000, page 5), R(x) = (log(x) + gamma + log(8)/2)/Pi, needs independent confirmation. gamma is A001620.

%H D. Atkinson and F. J. van Steenwijk, <a href="http://dx.doi.org/10.1119/1.19311">Infinite resistive lattices</a>, Am. J. Phys. 67 (1999), 486-492. (See A211074 for an alternative link.)

%H J. Cserti, <a href="http://arxiv.org/abs/cond-mat/9909120">Application of the lattice Green's function for calculating the resistance of infinite networks of resistors</a>, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.

%e a(0) = 1: R(1,0) = 1/2 is the first resistance > 0;

%e a(1) = 5: R(4,0) = 0.953987..., R(5,0) = 1.025804658...;

%e a(2) = 107: R(106,0) = 1.999103258858..., R(107,0) = 2.002092149977722...;

%e a(3) = 2460: R(2459,0) = 2.999894481..., R(2460,0) = 3.0000239019301...;

%e a(4) = 56922: R(56921,0) = 3.99999536602..., R(56922,0) =  4.0000009581... .

%o (PARI) \\ can be used to calculate estimates of terms for n >= 2, using the asymptotic formula. For n <= 8 results identical to those using the exact evaluation of the full integral are produced, but equality for higher terms might not hold, although with extremely remote probability.

%o a355955_asymp(upto) = {my(c=2.2, Rsqasy(L)=(1/Pi)*(log(L)+Euler+log(8)/2), d, m); for (n=2, upto, d=exp(c*n); d=solve(x=0.5*d, 2.5*d, Rsqasy(x)-n); print1(ceil(d),", "); c=log(d)/n)};

%o a355955_asymp(8)

%Y Cf. A355565, A355589 (same problem for triangular lattice).

%K nonn,more

%O 0,2

%A _Hugo Pfoertner_, Jul 23 2022