OFFSET
0,3
COMMENTS
See A355585 for more information.
LINKS
R. J. Mathar, Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice, viXra:2208.0111 (2022).
EXAMPLE
The triangle begins:
0;
0;
-2, 1;
-24, 5;
-280, 64, -14;
-3400, 808, -111;
-212538, 51929, -9054, 1989;
-2708944, 673429, -127303, 15576;
-244962336, 61623224, -12361214, 1891328, -405592;
-3195918288, 810930216, -169618717, 28113999, -3217136;
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};
jk(j, k) = {my(jj=j, kk=k); if(k<1, jj=j-k+1; kk=2-k); my(km=(jj+1)/2); if(kk>km, kk=2*km-kk); [jj, kk]};
D(n) = subst(pollegendre(n), 'x, 7);
uv(k) = (Rtri(k, 0) - 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};
UV(nend) = { my(nmax=nend+1, M=matrix(nmax, (nmax+1)\2)); for (n=3, nmax, M[n, 1] = bestappr(uv(n-1), poddpri(n-1))); for (n=3, nmax, M[n, 2]=(1/2)*(6*M[n-1, 1] - 2*M[jk(n-1, 2)[1], jk(n-1, 2)[2]] - M[n-2, 1] - M[n, 1])); for (n=5, nmax, for (m=3, (n+1)\2, M[n, m] = 6*M[jk(n-1, m-1)[1], jk(n-1, m-1)[2]] - M[jk(n-1, m)[1], jk(n-1, m)[2]] - M[jk(n-2, m-1)[1], jk(n-2, m-1)[2]] - M[jk(n-2, m-2)[1], jk(n-2, m-2)[2]] - M[jk(n-1, m-2)[1], jk(n-1, m-2)[2]] - M[jk(n, m-1)[1], jk(n, m-1)[2]] )); M};
UV(11)
CROSSREFS
KEYWORD
tabf,frac,sign
AUTHOR
Hugo Pfoertner, Jul 09 2022
STATUS
approved