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

Author

Hugo Pfoertner, Jul 23 2022

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

Links

• D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486-492. (See A211074 for an alternative link.)

Crossrefs

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