A176497 - OEIS

A176497

a(n) is the cardinality of the "Cross Set" which is the subset of distinct resistances that can be produced by a circuit of n unit resistors using only series or parallel combinations which cannot be decomposed as a single unit resistor in either series or parallel with a circuit of n-1 unit resistors.

%I #36 Dec 09 2024 05:30:18

%S 0,0,0,1,4,9,25,75,195,475,1265,3135,7983,19697,50003,126163,317629,

%T 802945,2035619,5158039,13084381,33240845,84478199,214717585,

%U 546235003,1389896683,3537930077,9007910913,22942258567,58444273501

%N a(n) is the cardinality of the "Cross Set" which is the subset of distinct resistances that can be produced by a circuit of n unit resistors using only series or parallel combinations which cannot be decomposed as a single unit resistor in either series or parallel with a circuit of n-1 unit resistors.

%C 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.

%C 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.

%H Antoni Amengual, <a href="http://dx.doi.org/10.1119/1.19396">The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel</a>, American Journal of Physics, 68(2), 175-179 (February 2000).

%H Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1004.3346/">The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel</a>, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).

%H Hugo Pfoertner, <a href="/plot2a?name1=A176497&name2=A048211&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true&drawlines=true">Plot of a(n)/A048211(n) vs n</a>, using Plot 2.

%F a(n) = A048211(n) - 2*A048211(n-1).

%e A(1) has no cross set and the first term is defined to be zero; the cross sets for n = 2 and n = 3 are empty hence the second and third term are each zero. Noting that A(3) = 4 and A(4) = 9, the fourth term is 1. The fifth term is 4.

%Y Cf. A048211, A176498, A176498.

%Y Cf. A153588, A174283, A174284, A174285 and A174286, A176499, A176500, A176501, A176502. [_Sameen Ahmed Khan_, May 02 2010]

%K more,nonn

%O 1,5

%A _Sameen Ahmed Khan_, Apr 21 2010

%E a(23) from _Sameen Ahmed Khan_, May 02 2010

%E a(24)-a(25) from _Antoine Mathys_, Mar 19 2017

%E a(26)-a(30) from _Antoine Mathys_, Dec 08 2024

%E Edited by _Andrew Howroyd_, Dec 08 2024