cp's OEIS Frontend

A355951 Negated column 0 of the irregular triangle A355587.

%I A355951 #9 Aug 13 2022 15:46:01
%S A355951 0,0,2,24,280,3400,212538,2708944,244962336,3195918288,42013225014,
%T A355951 111125508824,11603576403816,30966112647080,188641282541015866,
%U A355951 2532986569522773024,34096877865475065728,459984329860282638816,105694712757690117569946,1431044069320995796765272,73738714208458783084303688
%N A355951 Negated column 0 of the irregular triangle A355587.
%C A355951 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 A355951 R(n) > 1 [ohm] for n >= 38. Cserti (2000, page 11) claims that R(n) is logarithmically divergent for large values of n.
%H A355951 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 A355951 (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 A355951 D(n) = subst(pollegendre(n), x, 7);
%o A355951 uv(k) = (Rtri(k) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
%o A355951 poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
%o A355951 for (k=0, 20, print1(-numerator(bestappr(uv(k),poddpri(k))), ", "))
%o A355951 \\ for A355952 replace by
%o A355951 \\ for (k=0, 20, print1(denominator(bestappr(uv(k),poddpri(k))),", "))
%Y A355951 Cf. A355587, A355952 (denominators).
%Y A355951 Cf. A084768, A355585, A355586, A355588.
%K A355951 nonn,frac
%O A355951 0,3
%A A355951 _Hugo Pfoertner_, Jul 21 2022