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%I #9 Aug 13 2022 15:46:01
%S 0,0,2,24,280,3400,212538,2708944,244962336,3195918288,42013225014,
%T 111125508824,11603576403816,30966112647080,188641282541015866,
%U 2532986569522773024,34096877865475065728,459984329860282638816,105694712757690117569946,1431044069320995796765272,73738714208458783084303688
%N Negated column 0 of the irregular triangle A355587.
%C a(n) are the numerators u in the representation R = s/t - (2*sqrt(3)/Pi)*u/v of the resistance between two nodes with distance n on the same grid line in an infinite triangular lattice of one-ohm resistors. The corresponding denominators are A355952. s(n)/t(n) = (1/3)*Sum_{k=0..n-1} A084768(k-1) for n >= 0.
%C R(n) > 1 [ohm] for n >= 38. Cserti (2000, page 11) claims that R(n) is logarithmically divergent for large values of n.
%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.
%o (PARI) Rtri(n, p) = {my(alphat(beta)=acosh(2/cos(beta)-cos(beta))); intnum (beta=0, Pi/2, (1 - exp (-abs(n-p) * alphat(beta))*cos((n+p)*beta)) / (cos(beta)*sinh(alphat(beta)))) / Pi};
%o D(n) = subst(pollegendre(n), x, 7);
%o uv(k) = (Rtri(k) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
%o poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
%o for (k=0, 20, print1(-numerator(bestappr(uv(k),poddpri(k))), ", "))
%o \\ for A355952 replace by
%o \\ for (k=0, 20, print1(denominator(bestappr(uv(k),poddpri(k))),", "))
%Y Cf. A355587, A355952 (denominators).
%Y Cf. A084768, A355585, A355586, A355588.
%K nonn,frac
%O 0,3
%A _Hugo Pfoertner_, Jul 21 2022
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