cp's OEIS Frontend

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.

%I A355587 #16 Sep 19 2022 14:03:15
%S A355587 0,0,-2,1,-24,5,-280,64,-14,-3400,808,-111,-212538,51929,-9054,1989,
%T A355587 -2708944,673429,-127303,15576,-244962336,61623224,-12361214,1891328,
%U A355587 -405592,-3195918288,810930216,-169618717,28113999,-3217136,-42013225014,2146081719,-2315951182,81986531,-57942922,12257507
%N 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.
%C A355587 See A355585 for more information.
%H A355587 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 A355587 The triangle begins:
%e A355587             0;
%e A355587             0;
%e A355587            -2,         1;
%e A355587           -24,         5;
%e A355587          -280,        64,        -14;
%e A355587         -3400,       808,       -111;
%e A355587       -212538,     51929,      -9054,     1989;
%e A355587      -2708944,    673429,    -127303,    15576;
%e A355587    -244962336,  61623224,  -12361214,  1891328,  -405592;
%e A355587   -3195918288, 810930216, -169618717, 28113999, -3217136;
%o A355587 (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 A355587 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 A355587 D(n) = subst(pollegendre(n), 'x, 7);
%o A355587 uv(k) = (Rtri(k,0) - sum(j=0, k-1, D(j))/3) / (2*sqrt(3)/Pi);
%o A355587 poddpri(primax) = {my(pp=1,p=2); while (p<=primax, p=nextprime(p+1); pp*=p); pp};
%o A355587 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 A355587 UV(11)
%Y A355587 A355588 are the corresponding denominators v.
%Y A355587 A355585 and A355586 are s and t.
%K A355587 tabf,frac,sign
%O A355587 0,3
%A A355587 _Hugo Pfoertner_, Jul 09 2022