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

0,3

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

Table of n, a(n) for n=0..20.

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.

Prog

(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))), ", "))

Crossrefs

Cf. A355587, A355952 (denominators).

Cf. A084768, A355585, A355586, A355588.

Sequence in context: A221082 A002006 A230129 * A065101 A052739 A135389

Adjacent sequences: A355948 A355949 A355950 * A355952 A355953 A355954

Keyword

nonn,frac

Author

Hugo Pfoertner, Jul 21 2022

Status

approved