A355951 Negated column 0 of the irregular triangle A355587.

0, 0, 2, 24, 280, 3400, 212538, 2708944, 244962336, 3195918288, 42013225014, 111125508824, 11603576403816, 30966112647080, 188641282541015866, 2532986569522773024, 34096877865475065728, 459984329860282638816, 105694712757690117569946, 1431044069320995796765272, 73738714208458783084303688

Offset: 0

Author

Hugo Pfoertner, Jul 21 2022

Keywords

• nonn
• frac

Comments

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.

R(n) > 1 [ohm] for n >= 38. Cserti (2000, page 11) claims that R(n) is logarithmically divergent for large values of n.

Links

• J. Cserti, Application of the lattice Green's function for calculating the resistance of infinite networks of resistors, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.

Crossrefs

Cf. A355587, A355952 (denominators).

Cf. A084768, A355585, A355586, A355588.

Programs

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};
D(n) = subst(pollegendre(n), x, 7);
uv(k) = (Rtri(k) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
for (k=0, 20, print1(-numerator(bestappr(uv(k),poddpri(k))), ", "))
\\ for A355952 replace by
\\ for (k=0, 20, print1(denominator(bestappr(uv(k),poddpri(k))),", "))