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.

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

Offset: 0

Author

Sameen Ahmed Khan, Mar 15 2010

Keywords

• more
• nonn

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

Examples

			Since a bridge circuit requires a minimum of five resistors, 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]
		

Links

• 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), 175-179 (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.gen-ph], (20 April 2010).
• Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162.
• Sameen Ahmed Khan, Beginning to Count the Number of Equivalent Resistances, Indian Journal of Science and Technology, Vol. 9, Issue 44, pp. 1-7, 2016.
• Rainer Rosenthal, Maple program SetA174283 used for A174284
• Index to sequences related to resistances.

Crossrefs

Cf. A048211, A153588, A174283, A174285, A174286, A176499, A176500, A176501, A176502, A180414.

Programs

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(n-1), SetA174283(n)) fi end:
A174284 := n -> nops(AccumulatedSetsA174283(n)):
seq(A174284(n), n=1..9); # Rainer Rosenthal, Oct 27 2020

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

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