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
Examples
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.
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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 ---
|
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With a numbering of the resistors as shown in the following diagram,
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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____|
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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].
Links
- Hugo Pfoertner, Table of n, a(n) for n = 1..50
- Hugo Pfoertner, Graph of R(n), bounded or unbounded for n->oo?
Programs
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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
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