

A174284


Number of distinct finite resistances that can be produced using at most n equal resistors (n or fewer resistors) in series, parallel and/or bridge configurations.


14



0, 1, 3, 7, 15, 35, 79, 193, 493, 1299, 3429, 9049, 23699, 62271, 163997, 433433, 1147659, 3040899
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OFFSET

0,3


COMMENTS

This sequence is a variation on A153588, which uses only series and parallel combinations. The circuits with exactly n unit resistors are counted by A174283, so this sequence counts the union of the sets, which are counted by A174283(k), k <= n.  Rainer Rosenthal, Oct 27 2020
For n = 0 the resistance is infinite, therefore the number of finite resistances is a(0) = 0. Sequence A180414 counts all resistances (including infinity) and so has A180414(0) = 1 and A180414(n) = a(n) + 1 for all n up to n = 7. For n > 7 the networks get more complex, producing more resistance values, so A180414(n) > a(n) + 1.  Rainer Rosenthal, Feb 13 2021


LINKS

Table of n, a(n) for n=0..17.
Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175179 (February 2000).
Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.genph], (20 April 2010).
Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153162.
Sameen Ahmed Khan, Beginning to Count the Number of Equivalent Resistances, Indian Journal of Science and Technology, Vol. 9, Issue 44, pp. 17, 2016.
Rainer Rosenthal, Maple program SetA174283 used for A174284
Index to sequences related to resistances.


FORMULA

a(n) = #(union of all S(k), k <= n), where S(k) is the set which is counted by A174283(k).  Rainer Rosenthal, Oct 27 2020


EXAMPLE

Since a bridge circuit requires a minimum of five resistances, the first four terms coincide with A153588. The fifth term also coincides since the set corresponding to five resistors for the bridge, i.e. {1}, is already obtained in the fourth set corresponding to the fourth term in A153588. [Edited by Rainer Rosenthal, Oct 27 2020]


MAPLE

# SetA174283(n) is the set of resistances counted by A174283(n) (see Maple link).
AccumulatedSetsA174283 := proc(n) option remember;
if n=1 then {1} else `union`(AccumulatedSetsA174283(n1), SetA174283(n)) fi end:
A174284 := n > nops(AccumulatedSetsA174283(n)):
seq(A174284(n), n=1..9); # Rainer Rosenthal, Oct 27 2020


CROSSREFS

Cf. A048211, A153588, A174283, A174285, A174286, A176499, A176500, A176501, A176502, A180414.
Sequence in context: A077946 A077970 A338852 * A182892 A124696 A081669
Adjacent sequences: A174281 A174282 A174283 * A174285 A174286 A174287


KEYWORD

more,nonn,changed


AUTHOR

Sameen Ahmed Khan, Mar 15 2010


EXTENSIONS

a(8) corrected, a(9)a(17) from Rainer Rosenthal, Oct 27 2020
Title changed and a(0) added by Rainer Rosenthal, Feb 13 2021


STATUS

approved



