

A355955


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


5




OFFSET

0,2


COMMENTS

The terms are obtained by a highprecision 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(m1,0)) < 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.
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.


LINKS



EXAMPLE

a(0) = 1: R(1,0) = 1/2 is the first resistance > 0;
a(1) = 5: R(4,0) = 0.953987..., R(5,0) = 1.025804658...;
a(2) = 107: R(106,0) = 1.999103258858..., R(107,0) = 2.002092149977722...;
a(3) = 2460: R(2459,0) = 2.999894481..., R(2460,0) = 3.0000239019301...;
a(4) = 56922: R(56921,0) = 3.99999536602..., R(56922,0) = 4.0000009581... .


PROG

(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.
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)};
a355955_asymp(8)


CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



