[seqfan] Re: Question regarding A174283

[M. F. Hasler]

On Tue, Oct 13, 2020 at 6:01 PM Rainer Rosenthal <r.rosenthal at web.de> wrote:

> thanks to Hugo Pfoertner we now have the missing resistance. Its value is
> 1/2.


(resulting from a 6 arm bridge - as I expected!)

So it is neither in the A048211(6) = 53 resistances {1, 3, 6, 3/2, ...10/9,
> 11/10, 10/11}
> nor in the A174285(6) = 3 resistances {11/13, 1, 13/11},

thus completing the A174283(6) = 57 resistances.
>


> I suggest to show the 53 resistances as a not so trivial example a(6) in
> addition to a(2) in A048211.
> The example a(6) = 57 in A174283 can then easily given in short form as
> A048211(6) + A174285(6) + 1.
> The additional resistance 1/2 is the resistance of the following "6-arm"
> bridge:(*snip*)
>

Yes, I completely agree that this would be a good idea!
These examples needed editing, anyway (which I had already started a tiny
bit).

And in addition to the sequence counting "more than 5 arm bridge"
contributions,
even though this case does maybe not occur up to n = 6,
there could be a sequence counting the number of same values produced by
different constructions, in particular when a "bridge" gives the same value
as a non-bridge circuit. (It would also be interesting to know whether
different serial / parallel circuits can give the same value, making a
hypothetical "naive" formula for these cases only an upper limit for
A048211.)


- Maximilian


> Am 12.10.2020 um 16:59 schrieb M. F. Hasler:
> > A174285 says "...confined to 5 arms...".
> > So the additional one, counted in A174283(6), probably has more than
> 5 arms...
> >
> > Also, the idea of  A174283 = A048211 + A174285 will go wrong
> > (in the other sense) when the two distinct constructions yield common
> values, viz
> > # ( A ∪ B ) = # A + # B  −  # ( A ∩ B ).
>
>