OFFSET
0,2
COMMENTS
The terms are obtained by a high-precision evaluation of the integral R(j,k) = (1/Pi) * Integral_{y=0..Pi/2} (1 - exp(-|j-k|*x)*cos((j+k)*y)) / (sinh(x)*cos(y)) dy, with x = arccosh(2/cos(y)-cos(y)), 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.
LINKS
D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486-492. (See A211074 for an alternative link.)
EXAMPLE
a(0) = 1: R(1,0) = 1/3 is the first resistance > 0;
a(1) = 38: R(37,0) = 0.9980131561985..., R(38,0) = 1.0029141482654...;
a(2) = 8632: R(8631) = 1.99999787859849..., R(8632) = 2.000019169949784851...;
a(3) = 1991753: R(1991752) = 2.99999998586..., R(1991753) = 3.000000078131...;
a(4) = 459625866: R(459625865)=3.999999999731...; R(459625866)=4.000000000131....
Assuming a fitted asymptotic logarithmic growth of R(x,0) = log(x)/(Pi*sqrt(3)) + 0.334412..., a(5) is approximately 1.06*10^11, but 250 GByte of main memory is not enough for PARI's function intnum to compute the value of the integral for arguments of that size.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Hugo Pfoertner, Jul 23 2022
STATUS
approved