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This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176502 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, Jul 12 2019]

Farey(n) = A005728(n). - Franklin T. Adams-Watters, May 12 2010

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 11 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 24 2021

This sequence arises in the analytically obtained strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. This sequence provides a better strict upper bound than A176499 but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]

From Hugo Pfoertner, Jan 24 2021: (Start)

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is wrong. It only applies to purely serial-parallel networks, but it already fails when bridges are allowed, as described in A174283. Even more so if arbitrary nonplanar networks are allowed as in A337517. See the linked illustrations of the respective quotients.

But in contrast to A176499, which at least correctly bounds A048211, the terms a(5), ..., a(9) in this sequence are smaller than the corresponding terms from A048211 (a(n) vs. A048211(n): 19/22, 50/53, 122/131, 317/337, 837/869). (End)

Each network with 2, 3 or 4 resistors is made up of series or parallel connected resistors in such a way that the resulting resistances can be computed as Ser(x1,x2) = x1 + x2 (type S) or Par(x1,x2) = 1/(1/x1+1/x2) (type P). The parameters are either 1 Ohm or themselves of type S or P. A048211 counts the different resistances which can be produced as S or P type from n unit resistors. With 5 resistors x1 .. x5 there is the bridge configuration (type B),

A which cannot be computed by functions Ser() and Par().

/ \ The resistance between A and D is given by

x1 x2

/ \ Bri(x1,x2,x3,x4,x5) =

B- x3 - C

\ / x2*x1*x4+x2*x1*x5+x5*x4*x1+x5*x4*x2+x3*(x2+x5)*(x1+x4)

x4 x5 ------------------------------------------------------ .

\ / (x1+x2)*(x4+x5)+x3*(x1+x4+x2+x5)

D

Sequence A174283 counts all resistances of types S, P and B which can be produced with n unit resistors. The next essentially new figuration comes with 7 resistors: the fork (type F), which cannot be computed by functions Ser(), Par() and Bri().

A

/ \

x3 x1

/ \

B- x5 - C

/ \ /

x4 x7 x6

/ \ /

E- x2 - D

The resistance between A and E is given by

Frk(x1,x2,x3,x4,x5,x6,x7) =

x1*x3*x4*x7+x1*x3*x4*x5+x1*x3*x2*x7+x1*x3*x2*x5+x2*x4*x3*x7+x2*x4*x3*x5+

x2*x4*x1*x7+x2*x4*x1*x5+x5*x7*x1*x3+x5*x7*x1*x4+x5*x7*x2*x3+x5*x7*x2*x4+

x6*x1*x3*x7+x6*x1*x3*x2+x6*x1*x3*x4+x6*x5*x7*x3+x6*x5*x2*x3+x6*x3*x4*x5+

x6*x3*x4*x7+x6*x1*x4*x7+x6*x5*x7*x4+x6*x2*x4*x3+x6*x2*x4*x1+x6*x5*x2*x4

------------------------------------------------------------------------ .

x3*x4*x7+x3*x4*x5+x2*x3*x7+x5*x2*x3+x1*x4*x7+x5*x1*x4+x1*x2*x7+

x1*x2*x5+x5*x7*x3+x5*x7*x4+x5*x7*x1+x5*x7*x2+x6*x3*x7+x6*x2*x3+

x6*x3*x4+x6*x1*x7+x6*x1*x2+x6*x1*x4+x6*x5*x7+x6*x5*x2+x6*x4*x5

This sequence A337516 counts all resistances of type S, P, B or F which can be produced with n unit resistors.

Karnofsky (2004, p. 5): "[...] if some circuit has resistance m/n then some other circuit likely has n/m. In fact, for 9 or fewer resistors, this symmetry is perfect. However, for 10 resistors the following values are achieved, but not their inverses: 95/106, 101/109, 98/103, 97/98, 103/101, 97/86, 110/91, 103/83, 130/101, 103/80, 115/89, 106/77, 109/77, 98/67, 101/67". That means, that T(m,n) = T(n,m), if T(m,n) <= 9.

This starts with the values of A113881, but the Karnofsky comment says that T(n,m) is not symmetric, whereas the count of tiles in A113881 is. - R. J. Mathar, Nov 06 2020

The first difference where T(m,n) = T(n,m), but differs from the corresponding entry of A113881 occurs for (n,m) = (154,167) and (n,m) = (167,154), both representable by networks with non-planar graphs of 11 resistors, whereas A113881 counts 12 tiles. See Pfoertner link for illustration of more differences. - Hugo Pfoertner, Nov 13 2020

A connected multigraph G with a selected pair P of nodes can be used to represent a resistor network. The edges represent resistors, and the total resistance is measured between the selected nodes. It is possible to construct complex networks using only serial or parallel combinations, but the more nodes and edges are involved, the more networks of a different kind can be found. They cannot be decomposed into serial/parallel elements. The sequence is on page 2 of the paper describing the computation of A180414 (see the Joel Karnofsky link).

Karnofsky claims that he systematically increased the number of edges by three basic operations, C, D, and E, defined in A338999, i.e., he claims to have counted the CDE-descendants of the simplest h-graph (the "bridge," see the example section). Numbers given in his paper are 1, 5, 37, 226, 1460, 9235, which is slightly off (see A339386). The difference seems to stem from the "dangling parts," as he calls them in his "addendum," so they don't affect the computation of different resistances in A180414. - Rainer Rosenthal, Dec 02 2020

This sequence arises in the decomposition of the sets A(n + 1) of equivalent resistances, when n equal resistors are combined in series/parallel, into series parallel and cross sets respectively. The order of the set A(n) of equivalent resistances when n resistors are combined in series/parallel is given by the Sequence A048211: 1, 2, 4, 9, 22, 53, 131, 337, 869, ... Treating the elements of A(n) as single blocks the (n + 1)th resistor can be added either in series or in parallel.

We call these two sets as series set and parallel set respectively. One can also add the (n + 1)th resistor somewhere within the A(n) blocks, and we call this set as the cross set. The series and the parallel sets each have exactly A(n) number of configurations and the same number of equivalent resistances. All the elements of the parallel set are strictly less than 1 and that of the series set are strictly greater than 1. These two disjoint sets contribute 2*A(n) number of elements to A(n + 1) and are the source of 2n. It is the cross set which takes the count beyond 2^n to 2.53^n numerically (up to n = 22) and maximally to 2.61^n, strictly fixed by the Farey scheme. The cross set is not straightforward, as it is generated by placing the (n + 1)th resistor anywhere within the blocks of A(n). The order of the cross set is A(n + 1) - 2*A(n) leading to this sequence.

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

Cumulative sequence based on A337516.