A355589 a(n) is the least distance of two nodes on the same grid line in an infinite triangular lattice of one-ohm resistors for which the resistance measured between the two nodes is greater than n ohms.

1, 38, 8632, 1991753, 459625866

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

Table of n, a(n) for n=0..4.

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

Cf. A355585, A355955 (same problem for square lattice).

Sequence in context: A173133 A221354 A096558 * A145120 A183581 A030260

Adjacent sequences: A355586 A355587 A355588 * A355590 A355591 A355592

Keyword

nonn,hard,more

Author

Hugo Pfoertner, Jul 23 2022

Status

approved