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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)

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 relation of this sequence to A340920 is the analog of the relation of A180414 to A337517.

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