%I #16 Sep 19 2022 14:03:15

%S 0,0,-2,1,-24,5,-280,64,-14,-3400,808,-111,-212538,51929,-9054,1989,

%T -2708944,673429,-127303,15576,-244962336,61623224,-12361214,1891328,

%U -405592,-3195918288,810930216,-169618717,28113999,-3217136,-42013225014,2146081719,-2315951182,81986531,-57942922,12257507

%N 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.

%C See A355585 for more information.

%H R. J. Mathar, <a href="http://vixra.org/abs/2208.0111">Recurrence for the Atkinson-Steenwijk Integrals for Resistors in the Infinite Triangular Lattice</a>, viXra:2208.0111 (2022).

%e The triangle begins:

%e 0;

%e 0;

%e -2, 1;

%e -24, 5;

%e -280, 64, -14;

%e -3400, 808, -111;

%e -212538, 51929, -9054, 1989;

%e -2708944, 673429, -127303, 15576;

%e -244962336, 61623224, -12361214, 1891328, -405592;

%e -3195918288, 810930216, -169618717, 28113999, -3217136;

%o (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 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]};

%o D(n) = subst(pollegendre(n), 'x, 7);

%o uv(k) = (Rtri(k,0) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);

%o poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};

%o 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};

%o UV(11)

%Y A355588 are the corresponding denominators v.

%Y A355585 and A355586 are s and t.

%K tabf,frac,sign

%O 0,3

%A _Hugo Pfoertner_, Jul 09 2022