A355587 T(j,k) are the numerators u in the representation R = s/t + (2*sqrt(3)/Pi)*u/v of the resistance between two nodes separated by the distance (j,k) in an infinite triangular lattice of one-ohm resistors, where T(j,k), j >= 0, 0 <= k <= floor(j/2) is an irregular triangle read by rows.

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, -42013225014, 2146081719, -2315951182, 81986531, -57942922, 12257507

Offset

0,3

Comments

See A355585 for more information.

Links

Table of n, a(n) for n=0..35.

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

A355588 are the corresponding denominators v.

A355585 and A355586 are s and t.

Sequence in context: A261407 A037943 A073876 * A276399 A119828 A328826

Adjacent sequences: A355584 A355585 A355586 * A355588 A355589 A355590

Keyword

tabf,frac,sign

Author

Hugo Pfoertner, Jul 09 2022

Status

approved