A338401 a(n) is the numerator of the resistance R(n) = a(n)/A338402(n) of a triangular network of 3*n*(n+1)/2 one Ohm resistors in a hexagonal lattice arrangement.

2, 10, 10, 206, 3326, 43118, 150806, 11591578, 436494606, 1008712015454, 382034633808890, 13187511533010430, 2111825680430510462, 171204772756285452656378, 89579048665281690355286, 1013412795315891086553473628734, 20023655015717377508089133638478, 24678955315461926144059519221489609194

Offset

1,1

Comments

The resistance is measured between two corners of the triangular region.

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..50

Hugo Pfoertner, Graph of R(n), bounded or unbounded for n->oo?

Example

R(1) = a(1)/A338402(1) = 2/3,
R(2) = a(2)/A338402(2) = 10/9,
R(4) = a(4)/A338402(4) = 206/123.
a(3) = 10: The following network of A045943(3) = 18 one Ohm resistors has a resistance of R(3) = 10/7 Ohm, i.e., the current I driven by the voltage of 1 Volt is 7/10 = A338402(3)/a(3) Ampere.
.
                       O
                    __/ \_
                   / /   \ \
                  /1/     \1\
                 /_/       \_\
                 /   _____   \
                O---|__1__|---O
             __/ \_        __/ \_
            / /   \ \     / /   \ \
           /1/     \1\   /1/     \1\
          /_/       \_\ /_/       \_\
          /   _____   \ /   _____   \
         O---|__1__|---O---|__1__|---O
      __/ \__       __/ \__       __/ \_
     / /   \ \     / /   \ \     / /   \ \
    /1/     \1\   /1/     \1\   /1/     \1\
   /_/       \_\ /_/       \_\ /_/       \_\
   /   _____   \ /   _____   \ /   _____   \
  O---|__1__|---O---|__1__|---O---|__1__|---O
  |                                         |
  |                 V = 1 Volt              |
  |                     |                   |
   -------------------| |-- I=1/R Ampere ---
                        |
.
With a numbering of the resistors as shown in the following diagram,
.
              O
             / \
           15  18
           /     \
          O--14---O
         / \     / \
        7   9  13  17
       /     \ /     \
      O-- 6---O--12---O
     / \     / \     / \
    2   3   5   8  11  16
   /     \ /     \ /     \
  O---1---O---4---O--10---O
  |______1 Volt__I=I19____|
.
the currents in Amperes through the 18 resistors, and the current I=I19 through the voltage source of 1 Volt, are [11/30, 1/3, 1/30, 4/15, 2/15, 1/6, 2/15, 2/15, 1/30, 11/30, 1/30, 1/6, 1/30, 1/15, 1/30, 1/3, 2/15, 1/30, 7/10].

Prog

(PARI) a33840_1_2(n)={my(md=3*n*(n+1)/2+1,
T1=matrix(n, n), T2=matrix(n, n), T3=matrix(n, n),
M=matrix(md, md, i, j, 0), U=vector(md),
valid(i, j)=i>0&&i<=n&&j>0&&j<=n&&i>=j, k=0, neq=1);
\\ List of edges
for(i=1, n, for(j=1, i, T1[i, j]=k++; T2[i, j]=k++; T3[i, j]=k++));
\\ In- and outflow of current at all nodes
\\ lower left triangle with inflow of current from source of voltage
M[1, 1]=-1; M[1, 2]=-1; M[1, md]=1;
\\ loops over lower left corners of triangles
for(i=2, n+1, for(j=1, i,
\\ exclude node at top of triangle
if(j<n+1, neq++;
if(valid(i-1, j), M[neq, T1[i-1, j]]=1; M[neq, T3[i-1, j]]=1);
if(valid(i-1, j-1), M[neq, T2[i-1, j-1]]=1; M[neq, T3[i-1, j-1]]=-1);
if ( valid(i, j), M[neq, T1[i, j]]=-1; M[neq, T2[i, j]]=-1);
\\ lower right corner with current through voltage source
if ( i == n+1 && j == 1, M[neq, md] = -1)
)));
\\ sum of voltages around triangles with vertex above base
for( i = 1, n, for( j = 1, i,
neq++; M[neq, T1[i, j]] = 1; M[neq, T2[i, j]] = -1; M[neq, T3[i, j]] = -1 ));
\\ sum of voltages around triangles with vertex below base
for( i = 1, n-1, for( j = 1, i, neq ++;
M[neq, T3[i, j]] = 1; M[neq, T2[i+1, j]] = 1; M[neq, T1[i+1, j+1]] = -1 ));
\\ External voltage applied to lower corners of triangle
neq = neq++; for ( i = 1, n, M[neq, T1[i, 1]] = 1);
\\ Right side of equations; driving voltage 1 Volt
U[neq]=1; 1/matsolve(M, U~)[neq]};
for(n=1, 10, print1(numerator(a33840_1_2(n)), ", "))

Crossrefs

Cf. A000217, A045943, A048211, A174283, A338402.

Sequence in context: A351659 A033466 A193181 * A222638 A299982 A367638

Adjacent sequences: A338398 A338399 A338400 * A338402 A338403 A338404

Keyword

nonn,frac

Author

Hugo Pfoertner, Oct 24 2020

Status

approved